text:
\section{Introduction}

 Bessel, Jacobi, Legendre, $3j$- and $6j$-symbols in quantum mechanics and many other classical special functions are special cases of hypergeometric functions.
 In fact, many elementary and other important functions in mathematics and physics can be written in terms of hypergeometric functions, including the Euler beta and the gamma function.
They can also be described as solitons of special differential equations, classified by singularities and exponents of certain differential equations.
They have explicit integral and series representations, transformation and summation formulas, and other beautiful formulations relating various representations of hypergeometric functions. These functions are found and studied in combinatorics (cf. \cite{MR2343232, MR1429893}), Hilbert spaces as classical orthogonal polynomial bases (cf. \cite{koepf1996algorithms}), quantum physics in the form of harmonic analysis (cf. \cite{dattoli2010euler}), integrable systems of nonlinear differential equations as $q$-hypergeometric series using elliptic and theta-functions (cf. \cite{MR2506157, MR2044635}), and representation theory as matrix coefficients of Lie group representations (cf. \cite{MR1476496}).

The gamma function was constructed by Euler in an attempt to find an analytic continuation of the factorial function. It has a representation as an infinite integral and as a limit of a finite product, and it describes factorials in the special case with an integral domain.
The beta function is one of the classical Euler integrals and it can define binomial coefficients after a certain adjustment of indices.
It is a fundamental tool
to systems of holonomic equations (\cite{MR902936}), as well as a fundamental special function in engineering (cf. \cite{MR3750264, MR3735044, MR3797763}),
analysis (cf. \cite{MR1926172, MR951750, MR2004947}),
number theory (cf. \cite{MR3726238, MR3773936, MR3773937, MR3773940}),
combinatorics (cf. \cite{mansour2001gamma, MR1755751})
and mathematical physics (cf. \cite{MR3798008, MR3780488, MR3763251, MR3759662}).
Furthermore, the beta function describes important properties of the strong nuclear force. That is, the nuclear interactions of elementary particles modeled using $1$-dimensional strings rather than using zero-dimensional particles are precisely described by the Euler beta function as a model for the scattering amplitude (cf. \cite{MR2006979, MR2284609}).

One of the important properties about the beta function is its close connection to the binomial coefficients that follow from Ramanujan's master theorem (cf. \cite{MR0106147} or Appendix):
\begin{equation}\label{}
B (r, s)   =
 \frac{\mathit{\Gamma} (r) \mathit{\Gamma} (s)}{\mathit{\Gamma} (r+s)}
 =
 \frac{\pi \sin (\pi(r + s))}{\sin (\pi r) \sin (\pi s)} \frac{\mathit{\Gamma} (1 - r - s)}{\mathit{\Gamma} (1 - r) \mathit{\Gamma} (1 - s)}
 =
 \frac{\pi \sin (\pi(r + s))}{\sin (\pi r) \sin (\pi s)}
 { - r - s \choose - r }, \nonumber
\end{equation}
where the second equality holds only when $r,s\not\in \mathbb{Z}$,
and the third equality holds in the sense of holomorphic functions, i.e., if the equality holds on a dense open set, then it holds everywhere, including at the singularities, due to the identity principle.

Let $G$ be a Lie group and let $P$  be a parabolic subgroup.
In this article we introduce the Euler beta super-integral, generalizing the classical beta function:
\begin{equation}
B (r, s)   :=
 \int_0^1 t^{r - 1} (1 - t)^{s - 1} dt , \hspace{4mm}\mbox{ where }
 \text{Re}(r), \hspace{2mm}\text{Re}(s) >0,
\end{equation}
where one integrates over a parameter cycle over a Grassmannian $G/P$.

Our approach is based on the technique of Gelfand (cf. \cite{MR841131, MR888012, MR1127265, MR1121103}), who introduced a geometric interpretation for general hypergeometric functions as sections of tautological bundle $ \tau \to G_{k, n} $ over a complex Grassmannian manifold $ G_{k, n} $, i.e., see   \cite{MR1900449}. In particular, the classical beta function can be realized in terms of the geometry of the Grassmannian
 $ G_{2, 3}$, which is explained in Section~\ref{subsection:connection-grassmannian-G23-beta}.

 Although one could work in the category of complex differential geometry (and use a description of the fundamental bundles), we will work in the algebraic geometry
point of view as one can study Grassmannians in positive characteristic, which is very useful in arithmetic geometry and number theory.

We consider the super-Grassmannian $G_{2|1,3|2}$, which consists of $2|1$-dimensional super-subspaces in $3|2$-dimensional complex superspace (cf. see Sections~\ref{subsection:super-grassmannians}, \ref{subsection:properties-super-Grass}, and \ref{section:classical-super-beta-integral}).  We then construct a certain $1$-form $\omega$ whose coefficient is a product of powers of linear forms on the super-Grassmann, and integrate it over a $1$-cycle $\gamma$, giving us the beta super-integral $ B (s, p_1, p_2; \xi, \xi', \eta) $ (also see Theorem~\ref{thm:Euler-super-integral-relation}).
This integral that we construct in Section~\ref{section:classical-super-beta-integral} is one of the simplest possible generalizations of the classical Euler beta function, as one can add arbitrary many odd variables to the picture.

\begin{thrm}\label{thm:Euler-super-integral}
The super-integral of $\int_{\gamma}\omega$ on the super-Grassmannian $G_{2|1,3|2}$ is
\begin{equation}
\Phi(s,p_1,p_2; \xi, \xi', \eta)  = - (-x_{21})^{-s-1} \int_0^1 \int_{-\infty}^{\infty} u^s (1 - u + \eta \theta)^{p_1} (\xi + \xi' u + \theta)^{p_2} \theta \, d \theta \,  d u .
\end{equation}
\end{thrm}

One may extend Theorem~\ref{thm:Euler-super-integral} to the super-monodromy setting using super-differential equations and super-connections. This construction will be discussed in our sequel paper.

\begin{thrm}\label{thm:Euler-super-integral-relation}
The beta super-integral on the super-Grassmannian $G_{2|1,3|2}$ is
\begin{equation}
\Phi(s,p_1,p_2; \xi, \xi', \eta)  = - (-x_{21})^{-s-1}  B (s, p_1, p_2; \xi, \xi', \eta).
\end{equation}
\end{thrm}

We use the axiom of Berezin integration that $\int_{-\infty}^{\infty}\theta \: d\theta =1$ to obtain:

\begin{lem}\label{lem:Euler-super-integral-classical-reduction}
If $ p_2 = 0 $ and
$ \eta = 0 $ on $G_{2|1,3|2}$,
we have
\begin{equation}\label{eqn:reduction-classical-Euler-beta}
B (s, p_1, 0; \xi, \xi', 0)   =
 \int_0^1 \int_{-\infty}^{\infty}  u^s (1 - u)^{p_1}   \theta \: d\theta d u
 =
  \int_0^1  u^s (1 - u)^{p_1}  d u
    =   B (s + 1, p_1 + 1) ,
\end{equation}
with the right-hand side $ B $ being the classical Euler beta function.
\end{lem}

\begin{thrm}\label{thrm:super-beta-gamma-function}
The beta super-integral on the super-Grassmannian $G_{2|1,3|2}$ also has the representation
\begin{equation}
B (s, p_1, p_2; \xi, \xi', \eta)
=
\frac{\Gamma(s+1)\Gamma(p_1+1)}{\Gamma(p_1+s+2)}
\xi^{p_2}  \,
{}_2F_1 \left( \begin{matrix} -p_2 , s+1 \\ p_1 + s+2 \end{matrix} \, \Big| - \frac{\xi'}{\xi}  \right).
\end{equation}
\end{thrm}

\subsection{Summary of the sections}\label{subsection:summary}

Section~\ref{section:background} recalls classical topics in the literature. Section~\ref{subsection:grassmannians} gives constructions of Grassmannians using quotients of algebraic groups, and we discuss integration on them. In Section~\ref{subsection:connection-grassmannian-G23-beta}, we give a connection between the geometry of a certain Grassmannian and the Euler beta integral. In Section~\ref{subsection:properties-beta-function-classical}, properties of the Euler beta function are given, including properties of the integral of a particular differential form over a cycle.  We define and construct vector superspaces in Section~\ref{subsection:superspace}, discussing superpoints and the Grassmann algebra.
Super-Grassmannians are constructed and discussed in Section~\ref{subsection:super-grassmannians}, and in Section~\ref{subsection:properties-super-Grass}, we give basic properties of super-Grassmannians. Finally, in Section~\ref{subsection:change-of-variables}, we define change of variables in the super setting.

In Section~\ref{section:classical-super-beta-integral}, we give the general construction of the beta super-integral (also see Section~\ref{subsection:general-beta-integral}), including a thorough exploration of the special case of the super-Grassmannian $G_{2|1,3|2}$ in Section~\ref{subsection:special-case}, thus proving Theorem~\ref{thm:Euler-super-integral} and Theorem~\ref{thm:Euler-super-integral-relation}.
In Section~\ref{section:super-beta-integral-odd-variables}, we expand the beta super-integral with respect to the odd variables, thus proving Theorem~\ref{thrm:super-beta-gamma-function}.
We conclude with an Appendix giving a more thorough discussion of Ramanujan's master theorem.

\subsection{Acknowledgement}\label{section:acknowledgement}

The first author is partially supported by National Research Council in Washington D.C. M.S.I. thanks Jan Kochanowski University in Kielce, Poland for their hospitality, where the initial draft of this paper was produced.  M.Z. would like to thank M.S.I. for introducing him to super-geometry and super-analysis.

\section{Background}\label{section:background}

We begin with a discussion on Grassmannians.
\subsection{Grassmannians}\label{subsection:grassmannians}
Let $ G_{k, n} $ denote the complex Grassmannian variety of $ k $-dimensional linear subspaces in an $ n $-dimensional  vector space. The atlas of $ G_{k, n} $ consist of maps $ x : G_{k, n} \to \mathbb{C}^{k (n - k) } $ constructed by the mappings from $ G_{k, n} $ to the space of matrices $ M_{k \times n} (\mathbb{C}) $, with rank equal to $ k $. They form open subsets in the space of matrices $ M_{k \times n} (\mathbb{C}) $
		as we exclude closed sets
defined as the zero locus $ \det A_{i_1 i_2 \ldots i_k} = 0 $, where $ A_{i_1 i_2 \ldots i_k} $ is a minor whose columns have indices $ i_1 , i_2 , \ldots , i_k $.
Since $G_{k,n}$ are isomorphic up to a change-of-basis,
we will denote the open locus $\det A_{i_1 i_2 \ldots i_k} \not= 0 $ in $ M_{k \times n} (\mathbb{C})$  by
 $ S_{k, n} $.
The number of such (distinct) maps is equal to the number of choices of $ k $ columns from $ n $ columns, i.e., $ { n \choose k } $.
 The choice of these columns associates a natural (left) action of $ GL_k ( \mathbb{C})$ on $ S_{k, n} $ defined by taking  the invertible minor to the identity matrix.
Thus quotienting out $S_{k,n}$ by $ GL_k ( \mathbb{C})$, we obtain $ G_{k, n} = GL_k ( \mathbb{C}) \backslash S_{k, n} $.
We also have the torus $T_k := GL_1 ( \mathbb{C})^k \cong GL_1(\mathbb{C})^{\oplus k}\oplus 1^{\oplus (n-k)}$  acting on $S_{k, n}$
diagonally from the right.
Let us denote the possibly singular space of parameters
 by $ P_{k, n}  = GL_k (\mathbb{C}) \backslash S_{k, n} / T_k $.

For $1\leq i\leq k$,
let $ l_i : = x_{1i} + x_{2i} u_2 + \ldots + x_{ki} u_k$,
and let $ \phi_x (u) : = l_1^{s_1} l_2^{s_2} \cdots l_k^{s_k} $.
We define hyperplanes as $ L_j = \{ l_j = 0 \}$, and
divisors as $ D_x  = \bigcup_j L_j $.

Let $ s \in U \subset \mathbb{C}^k $ denote a complex parameter.
 Define
\begin{equation}\label{AomotoGelf}
\Phi (x, s)  :=
 \int_{\gamma_x} \phi_x (u) \, du,
\end{equation}
where
$\gamma_x$ is a cycle on $ \mathbb{CP}^{k-1} \backslash (D_x \cap \mathbb{RP}^{k-1} $).
Integrate over the cycle $ \gamma_x $, and homological with the cycle is a connected component of $ \mathbb{CP}^{k-1} \backslash (D_x \cap \mathbb{RP}^{k-1} $),
lying in-between (and thus bounded by) the hyperplanes  $L_j$. Thus $\gamma_x$ is also bounded by $D_x$.

The function $ \Phi $ is well-defined and it has an analytic continuation (along any curve) to the entire $ S_{k, n} $
by the identity principle.\footnote{The set of parameters is from $S_{k,n}$, and is invariant under the $GL_k(\mathbb{C})$-action, giving us $G_{k,n}$. Thus, we just integrate over $\mathbb{CP}^{k-1}$.
}
Furthermore, $ \Phi $ is \emph{invariant} with respect to the action of the special linear group $ SL_k ( \mathbb{C}) \subset GL_k (\mathbb{C}) $, which is evident in the following derivation. Define the left action by $GL_k (\mathbb{C})$ on $u$ by $g . u$, where $u=(u_1,\ldots, u_k)$.
First, if $ g \in GL_k (\mathbb{C}) $, then
\begin{align}
\Phi (g. x, s) & = \,
 \int_{\gamma_{g . x}} \phi_{g . x} (u) \, du \, = \,
 \int_{\gamma_{g . x}} \phi_{x} (g^{-1} u) \, du \nonumber \\ & = \,
 (\det g)^{-1}  \int_{\gamma_{g . x}} \phi_{x} (v) \, dv .
\end{align}
Since $ GL_k (\mathbb{C}) $ is a connected complex manifold, there exists a curve, say $ g (t) $, such that $ g (0) = I $ and $ g (1) = g $. It follows that  $ \gamma_{g . x} $ and $ \gamma_{x} $ are \emph{homologous}.
But if $ g \in SL_k (\mathbb{C}) $, then
\begin{equation}
\Phi (g. x, s) \, = \,
 \Phi (x, s) .
\end{equation}
But $ GL_1 (\mathbb{C}) \to GL_k (\mathbb{C}) \to SL_k (\mathbb{C}) \overset{\det}{\longrightarrow} 1 $ is a \emph{principal bundle},
 and it follows that $ \Phi $ is defined on the quotient space $ SL_k (\mathbb{C}) \backslash S_{k, n} $, which is a line bundle over $ G_{k, n} $.

Let $ \tau \to G_{k, n} $ be the \emph{tautological bundle}, parametrized by all pairs of the form $ (x, V_x)$, where $x \in G_{k, n}$ and $ V_x \subset \mathbb{C}^n $ is a linear subspace corresponding to $ x \in G_{k, n} $. The rank of $ \tau $ is equal to $ k $, which implies that $ \bigwedge^k \tau $ is a line bundle. The action of $ SL_k (\mathbb{C}) $ on $ \tau $  leads to the natural action on $ \bigwedge^k \tau $ on each section of $ \tau $, invariant with respect to this action.
This also produces an invariant section of $ \bigwedge^k \tau $.
But since such section is precisely $ \Phi $,
we will view $ \Phi $ as a \emph{section} of the line bundle $ \bigwedge^k \tau $.

\subsection{The geometry of the Grassmannian $G_{2,3}$ and the Euler beta function}\label{subsection:connection-grassmannian-G23-beta}

Let $ V_2 \subset \mathbb{C}^3 $ be a $2$-dimensional vector space. Then $ V_2 $ can be described using a basis of two linearly independent vectors from $ \mathbb{C}^3 $, which can be represented by the matrix
\begin{equation}\label{}
A  : =
 \left( \begin{array}{ccc} x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \end{array}
 \right)
\end{equation}
of maximal rank (equal to two). Because the rank is maximal, we can find an invertible minor, say
\begin{equation}\label{}
g \, : = \,
 \left( \begin{array}{cc} x_{11} & x_{12} \\ x_{21} & x_{22}
\end{array}
 \right) , \quad g \in GL_2 (\mathbb{C}) ,
\end{equation}
such that
\begin{equation}\label{}
g^{-1} A  : =
 \left( \begin{array}{ccc} 1 & 0 & y_1 \\ 0 & 1 & y_2 \end{array}
 \right) ,
\end{equation}
with $ y_1, y_2 $ given explicitly in terms of $ x_{ij} $.

We define an affine map $ \phi (g^{-1} A) = (y_1, y_2) \in \mathbb{C}^2 $ on the \emph{Grassmannian variety} $ G_{2, 3} $, which parametrizes $ 2 $-dimensional vector subspaces in $ \mathbb{C}^3 $.

\begin{rem}
From the other two minors, we can construct other maps that cover the entire $ G_{2, 3} $, and transitions between these maps provide the gluing conditions.  That is, for
\[
g' =
\begin{pmatrix}
x_{11} & x_{13} \\
x_{21} & x_{23} \\
\end{pmatrix}\in GL_2(\mathbb{C}),
\hspace{4mm}
(g')^{-1}A =
\begin{pmatrix}
1 & y_1' & 0 \\
0 & y_2' & 1 \\
\end{pmatrix}
\]
while for
\[
g''=
\begin{pmatrix}
x_{12} & x_{13} \\
x_{22} & x_{23} \\
\end{pmatrix}\in GL_2(\mathbb{C}),
\hspace{4mm}
(g'')^{-1}A =
\begin{pmatrix}
y_1'' & 1 & 0  \\
 y_2'' & 0 & 1 \\
\end{pmatrix}.
\]
\end{rem}

Now, let $ [w_1 : w_2] \in \mathbb{CP}^1 $. Then there are two maps of the form $ \phi :U\subseteq \mathbb{CP}^1 \to \mathbb{C} $, covering $ \mathbb{CP}^1 $. If we choose one of them, say $ w_1 \neq 0 $, then $ [w_1 : w_2] = [1 : t] $, with $ t \in \mathbb{C} $. From this data, we may construct three affine forms, i.e.,
the affine forms corresponding to $g^{-1}A$ are:
\begin{equation}\label{}
 \left( \begin{array}{c} 1 \\ t \end{array}
 \right)^T
 \left( \begin{array}{ccc} 1 & 0 & y_1 \\ 0 & 1 & y_2 \end{array}
 \right)   =
 \left( \begin{array}{ccc} 1 & t & y_1 + y_2 t \end{array}
 \right) .
\end{equation}
From these forms, we construct a function
\begin{equation}\label{eqn:reduction-three-products}
\phi (t)   =
 1 \cdot t^{\alpha} \cdot (y_1 + y_2 t)^{\beta} ,
\end{equation}
as well as the differential form $ \omega  = \phi (t) \, dt $.

To find the natural domain of integration for $ \omega $, note that the function $ \phi $ is multivalued and has singularities precisely when $ t = 0 $ or $ y_1 + y_2 t = 0 $. From the second equation, we deduce that the equality holds precisely when $ t = - y_1 / y_2 $. So $ \phi $, and thus also $ \omega $, is well-defined on the (universal cover of the) Riemann surface $ \mathbb{CP}^1 \backslash \{ 0, -y_1/y_2 , \infty \} $. Connecting these two points with a line gives the required
 chain of integration, say $ \gamma $.\footnote{We say $ \gamma $ is a (twisted) \emph{cycle} in the \emph{twisted homology}. } To find the other two chains of integration, we consider the other map $U\subseteq \mathbb{CP}^1 \to \mathbb{C} $ that covers $ \infty $, and this point is also singular. We thus obtain three singular points  $ 0 , - y_1/ y_2 , \infty $, and connecting them pairwise gives the three
 chains of integration.

\begin{defi}
Let $\phi$ be the map defined in \eqref{eqn:reduction-three-products}.
Define the function
\begin{equation}\label{eqn:one-cycle-one-form}
\Phi (\alpha, \beta) \, := \,
 \int_{\gamma} \omega .
\end{equation}
\end{defi}

\begin{defi}\label{defn:beta-integral}
For $\textrm{Re}(r), \textrm{Re}(s)>0$,
the {\em Euler beta integral} is
\begin{equation}\label{}
B (r, s) \, := \,
 \int_0^1 t^{r - 1} (1 - t)^{s - 1} \, dt .
\end{equation}
\end{defi}

\begin{rem}\label{rem:Euler-beta-classical-monodromy}
The Euler beta integral in Definition~\ref{defn:beta-integral} may be extended and defined for any complex numbers $r$ and $s$  using analytic continuation of differentiation by parts under the integral, using a technique known as monodromy.
\end{rem}

\subsection{Properties of the beta function}\label{subsection:properties-beta-function-classical}

We will describe several properties of $ \Phi $ in \eqref{eqn:one-cycle-one-form}, including its connection to the Euler beta integral.

\begin{thrm}\label{phi=beta}
Up to the constant $(- 1)^{\alpha+1} y_1^{\beta + \alpha + 1} y_2^{-\alpha - 1}$,
the function
$ \Phi (\alpha, \beta) $ is equal to
$ B (\alpha + 1, \beta + 1) $.
\end{thrm}

\begin{proof}
We have
\iffalse
\begin{equation}\label{}
\Phi (\alpha, \beta) \, = \,
 \int_0^{- y_1/y_2} t^{\alpha} (y_1 + y_2 t)^{\beta} \, dt \, = \,
 .
\end{equation}
\fi

\begin{align}\label{phibeta}
\Phi (\alpha, \beta) & =
 \int_0^{- y_1/y_2} t^{\alpha} (y_1 + y_2 t)^{\beta} \, dt \nonumber \\
 & =
 y_1^{\beta} \int_0^{- y_1/y_2} t^{\alpha} (1 + (y_2 / y_1) t)^{\beta} \, dt \nonumber \\
 & =
- y_1^{\beta} \int_0^1 (- (y_1 / y_2) u)^{\alpha} (1 - u)^{\beta} \, (y_1 / y_2) du \nonumber \\
 & =
 (- 1)^{\alpha+1} y_1^{\beta + \alpha + 1} y_2^{-\alpha - 1} \int_0^1 u^{\alpha} (1 - u)^{\beta} \, du  \nonumber \\
 & =
 (- 1)^{\alpha+1} y_1^{\beta + \alpha + 1} y_2^{-\alpha - 1} B (\alpha + 1, \beta + 1) ,
\end{align}
where we use the substitution $u=-(y_2/y_1)t$.
\end{proof}

\begin{rem}
The reduction
of $ \Phi $ to $ B $ resembles properties of $ \mathbb{CP}^1 $ with its complex structure. In addition, one could guess the formula of Theorem \ref{phi=beta} knowing that the dimension of the moduli space of complex structures on the Riemann sphere $ \mathbb{CP}^1 $ is $0-3 = -3$ since $ \mathrm{dim} \, PGL_2 (\mathbb{C}) = 3 $.\footnote{More precisely, this means that the set of complex structures of $ \mathbb{CP}^1 $ is just a point, but $ \mathbb{CP}^1 $ also has a $ 3 $-dimensional automorphism group $ PGL_2 (\mathbb{C}) \simeq GL_2 (\mathbb{C}) / \mathbb{C}^* $ that allows for transformation of three arbitrary points to, say, $ \{ 0, 1, \infty \} $.}

We give an alternative description, which is not as ``elegant''\footnote{This description is still elegant in some sense since the dimension of the moduli is always $ 3 g - 3 $, where $g$ is the genus.
}
but it is easier to comprehend: the dimension of moduli space is zero, and it admits a free action by $3$-dimensional group of automorphisms.
\end{rem}

\begin{rem}
Functions $ \Phi $ and $ B $ differ by a constant factor and thus are essentially the same, with the same qualitative properties. This can be understood in terms of a torus action that can be read in the proof of Theorem \ref{phi=beta}; in the second equality in the proof, we factor out the constant $y_1$
and then by a change of variable for the integral, we factor out $y_2$.
These procedures may be thought of as an adequate action by the torus $ \mathbb{C}^* \times \mathbb{C}^* $ on $ G_{2, 3} $.
\end{rem}

Permutation groups $ S_2 $ and $ S_3 $ act on the space $ G_{2, 3} $ by permuting the rows and the columns, respectively. The action of $ S_2 $ is not important as the basis elements for a $2$-dimensional subspace in a $3$-dimensional space produces the same $2$-dimensional space (regardless of the ordering on the basis vectors).
Furthermore, we can think of the $S_2$-action by permuting the rows of $G_{2,3}$ as restricting to the coordinate chart $\{[t:1] : t\in \mathbb{C} \}\subseteq \mathbb{CP}^1$, as opposed to the open set $\{ [1:t] : t\in \mathbb{C}\}$.\footnote{So the $S_2$-action can be thought of as moving between the charts, which is an additive operation.}

However, the action of $ S_3 $ is nontrivial.
Since $\int_{\gamma}\omega = \int_0^1 l_1^{s_1}l_2^{s_2}l_3^{s_3}dt$, the permutation group $S_3$ acts on the exponents $s_i$, where the transposition $(i,i+1)$ swaps the exponents $s_i$ and $s_{i+1}$.
Since $1^p = 1 $ for any $ p $, only the transposition $ (2, 3) $ acts on \eqref{eqn:reduction-three-products} in a nontrivial way.\footnote{There is a trivial factor $ 1^{\gamma} $ that is omitted, so the integrand reduces to $ 2 $ factors. Thus the biggest possible group that acts on the factors by permutation is $ S_2 $, and one checks by a straightforward computation that this is the subgroup of $ S_3 $ generated by the transposition $(2,3)$.}
This leads to the following result:

\begin{lem}
The action by the transposition $ (2, 3) $ on the space $ G_{2, 3} $ gives the formula
\begin{equation}\label{}
\Phi (\beta, \alpha) \, = \,
 ( - y_2)^{\alpha - \beta} \Phi (\alpha, \beta) .
\end{equation}
\end{lem}

\begin{proof}
We have
\begin{align}\label{}
\Phi (\beta, \alpha) & = \,
  (- 1)^{\beta+1} y_1^{\beta + \alpha + 1} y_2^{- \beta - 1} \int_0^1 u^{\beta} (1 - u)^{\alpha} \, du  \nonumber \\
 & = \,
  (- 1)^{\beta+1} y_1^{\beta + \alpha + 1} y_2^{- \beta - 1} \int_0^1 (1 - t)^{\beta} t^{\alpha} \, dt \nonumber \\
 & = \,
  (- 1)^{\beta - \alpha} y_2^{\alpha - \beta} \Phi (\alpha, \beta) \\
   & = \,
  (-y_2)^{\alpha - \beta} \Phi (\alpha, \beta) .
\end{align}
\end{proof}

Another property of $ \Phi $, as well as its more general counterpart, the hypergeometric function, is that it has close connection to a combinatorial function: the binomial coefficient
\begin{equation}
{ n \choose k } \, := \,
 \frac{n!}{(n - k)! k!} ,
\end{equation}
which can be shown in many ways, but we choose the one that can easily be generalized to other contexts. The main tool is \emph{Ramanujan's master theorem}, which we provide in more detail in the Appendix  (also see \cite{MR0106147}).

\begin{thrm}
The function $ \Phi $ is related to binomial coefficient by the equality
\begin{equation}\label{binom}
 \frac{\sin (\pi (\alpha+1))\sin (\pi(\beta+1))}{\pi \sin (\pi(\alpha+\beta+2))}
 (- 1)^{\alpha+1} y_1^{- \beta - \alpha - 1} y_2^{\alpha + 1} \Phi (\alpha, \beta) \, = \,
 { - \alpha - \beta-2 \choose - \alpha-1  }.
\end{equation}
\end{thrm}

\textit{Proof}. \, Applying the homographic change of variables and transforming the unit interval to positive half-line, we get
\begin{align}\label{}
(- 1)^{\alpha+1} y_1^{- \beta - \alpha - 1} y_2^{\alpha + 1} \Phi (\alpha, \beta) & = \,
  \int_0^1 u^{\alpha} (1 - u)^{\beta} \, du  \nonumber \\
 & = \,
 \int_0^{\infty} t (1 + t)^{- \alpha - \beta-2} t^{\alpha - 1} dt.
\end{align}
From the Taylor expansion of  $ (1 + t)^{- \alpha - \beta-2} $,
we have
\begin{equation}\label{eqn:power-series-expansion}
- t (1 + t)^{- \alpha - \beta-2}  =
 \sum_{n \geq 0} { - \alpha - \beta-2 \choose n } (-t)^{n + 1} ,
\end{equation}
and from Ramanujan's master theorem, we get (\ref{binom}). \, $ \square $

\subsection{Vector superspaces}\label{subsection:superspace}

Let $ (x_1, x_2, \ldots , x_n ; \xi_1, \ldots  , \xi_m) $ denote the standard variables on the superspace $ \mathbb{C}^{n|m} $; such a point is called a {\em superpoint}.
The basis of differential forms is given by $ (dx_1, dx_2, \ldots  , dx_n ; d\xi_1, \ldots  , d\xi_m) \in T^* \mathbb{C}^{n|m} $.
Since we have a natural $ n|m $-splitting on $ \mathbb{C}^{n|m} $, we first investigate an integral over an {\em odd point}, which involves only the odd coordinates, i.e., such a point lives on $\mathbb{C}^{0|m} $.

The integral satisfies basic rules such as linearity or shift-invariance. As functions on $ \mathbb{C}^{0|m} $ are precisely elements of the Grassmann algebra $ \mathbb{C} [\xi_1, \ldots , \xi_m] $, we would also require that, at least, the basis elements are integrable.

Let $ \int : \mathbb{C} [\xi_1, \ldots, \xi_m] \to \mathbb{C} $ be a linear functional satisfying linearity and shift-invariance. Then the anti-commutation relations $ \xi_i \xi_j = - \xi_j \xi_i $ imply that
$$
d\xi_i \wedge d\xi_j = -(-1)^{\bar{\xi_j}\bar{\xi_i}} d\xi_j \wedge d\xi_i = d\xi_j \wedge d\xi_i ,
$$
where
$\bar{\xi_i}\in \mathbb{Z}_2$ is the parity of $\xi_i$.
Thus
 $ \int d \xi_1 \wedge \cdots \wedge d\xi_m = 0$.\footnote{The translation invariance implies $ \int \varphi (\xi + \xi_0) d\xi = \int \varphi (\xi) d\xi $ for any constant $ \xi_0 $.
So letting $ \varphi(\xi) = \xi $, we get $ \xi_0 \int d\xi = 0 $; it follows that $\int d\xi = 0 $.}
But
\begin{equation}\label{eqn:nonzero-odd-integral}
 \int \xi_m \cdots \xi_1 \, d \xi_1 \wedge \cdots \wedge d\xi_m \neq 0 ,
 \end{equation}
 where the domain of integration does not need to be specified\footnote{It is a property of Berezin integral that the domain of integration is not specified, but rather, the integral should be algebraic and indefinite.}.
 As there is a freedom of scaling, we can assume that the integral \eqref{eqn:nonzero-odd-integral} is equal to one.\footnote{This assumption is natural since not only does it mimics the unit area of the unit hypercube, but also from the point of view of dual basis as we can read the relation $ \int \xi_i \: d \xi_j = \delta_{ij} $ as the duality condition. Moreover, as an operation on $ \mathbb{C} [\xi] $, it follows that the integral operation $ \int \cdot \, d \xi $ is equivalent to differentiation.}
  Then for any (polynomial) function $ \varphi : \mathbb{C}^{0| m} \to \mathbb{C} $, we have
\begin{equation}\label{}
\int_{\mathbb{C}^{0|m}}
 \varphi (\xi) d \xi \, = \,
 \varphi_{1, 2, \ldots , m} ,
\end{equation}
where $ \varphi_{1, 2, \ldots , m} $ is the `top' coefficient in the Taylor super-expansion:
\begin{equation}\label{}
\varphi (\xi) \, = \,
 \varphi_0 + \sum_{i = 1}^m \varphi_i \xi_i + \sum_{i < j}^m \varphi_{i, j} \xi_i \xi_j + \ldots + \varphi_{1, 2, \ldots , m} \xi_1 \xi_2 \cdots \xi_m,
\end{equation}
where the parity from the odd variables  has been incorporated in the coefficient of each monomial.

\subsection{Super-Grassmannians}\label{subsection:super-grassmannians}

An extensive treatment on supermanifolds and super-Grassmannians is found in \cite{MR1172996,MR3588978,witten2012notes}.

Let $ V^{k|l} \subset \mathbb{C}^{n|m} $ be a vector super-subspace of dimension $ k|l $. Then $ V^{k|l} $ can be described using a basis, i.e., a basis of $ k + l $ linearly independent $ \mathbb{Z}_2 $-graded vectors in $ \mathbb{C}^{n|m} $. But such basis can be put into a block matrix of the form
\begin{equation}\label{super-basis}
A \, = \,
 \left( \begin{array}{cc} x & \xi \\ \eta & y \end{array} \right)
\end{equation}
of maximal rank $ k + l $,
where $ x \in \mathbb{C}^k \otimes \mathbb{C}^n $, $ \xi \in \mathbb{C}^k \otimes \mathbb{C}^m $, $ \eta \in \mathbb{C}^l \otimes \mathbb{C}^n $ and $ y \in \mathbb{C}^l \otimes \mathbb{C}^m $. If $ U $ is a minor of $ A $ of maximal rank $ k + l $, then the basis corresponding to $ A $ is equivalent to the basis corresponding to $ U^{-1} A$, and all bases of $ V^{k|l} $ can be constructed in this way (for different minors). As $ U \in GL_{k|l} (\mathbb{C}) $, we are led to the following:

\begin{defi}[Super-Grassmannian]\label{defn:supergrassmannian}
Let $ B $ be the set of all bases of $ V^{k|l} $. The homogenous space $ G_{k|l , n|m} : = GL_{k|l} ( \mathbb{C})\backslash B $ is called the \emph{Grassmannian super-manifold}.
\end{defi}

We also refer to $G_{k|l , n|m}$ as the {\em super-Grassmannian}.

Another construction of the super-Grassmannian is via the unitary supergroup $ U (n|m) $. The unitary superbasis of the subspace $ V_{k|l} $ can be described as the
 unitary superbasis of $ \mathbb{C}^{n|m} $ modulo the action by $ U (n - k|m - l) $, which fixes $ n + m - k - l $ vectors, leaving the remaining $ k + l $ vectors arbitrary.

\begin{defi}[Stiefel super-manifold]
The homogenous space
$$ S_{k|l , n|m} : = U (n|m) / U (n - k|m - l)
$$
is called the \emph{Stiefel super-manifold}.
\end{defi}

In this super-unitary setting, the action of $ U (k|l) $ on the set of unitary basis of $ V^{k|l} $, i.e., on $ S_{k|l, n|m} $, corresponds to the action of $ GL_{k|l} (\mathbb{C}) $ on $ B $. Thus the definition of $ G_{k|l , n|m} $ is equivalent to the following:

\begin{defi}\label{defn:Grassmannian-super-mfd}
The homogenous space
$$
G_{k|l , n|m} : = U (n|m) / ( U (k|l) \times U (n - k|m - l))
$$
is called the \emph{Grassmannian super-manifold}.
\end{defi}

\begin{lem}
The two constructions of $ G_{k|l , n|m} $ in Definitions~\ref{defn:supergrassmannian} and \ref{defn:Grassmannian-super-mfd} are equivalent.
\end{lem}

\begin{proof}
The group $ U(n|m) $ is a deformation retract of $ GL_{n|m}(\mathbb{C}) $. This leads us to the same quotient space, with the use of a unitary vector in place of any  vector.
\end{proof}

In Section~\ref{subsection:properties-super-Grass}, we will use both constructions to explore the geometry and topology of Grassmannian super-manifolds.

\subsection{Properties of the super-Grassmannian}\label{subsection:properties-super-Grass}

As $ G_{k|l , n|m} $ is an algebraic supervariety, it can be described with a notion of \emph{duality}, i.e., by the sheaf $ \mathcal{O}_{G_{k|l , n|m}} $ of regular functions\footnote{The reconstruction of $G_{k|l , n|m}$
from $ \mathcal{O}_{G_{k|l , n|m}} $ is done with use of the theory of super-schemes, \textit{\`a la}  Grothendieck. This allows for further generalization, like inclusion of finite fields or, more generally, arbitrary super-commutative rings.}
 on $ G_{k|l , n|m} $. One can also construct an atlas which is
 super-algebraic, super-analytic, and super-smooth.

Let $ A $ be the matrix given in (\ref{super-basis}). It is a super-matrix of maximal rank so there exists a $ k|l \times k|l $ minor, say $ U $, such that $ U \in GL_{k|l}( \mathbb{C}) $. Then the rows of $ U^{-1} A $ also form a basis of the same subspace $ V^{k|l} \subset \mathbb{C}^{n|m} $. In this way we have an equivalence relation: $ A $ and $ B $ are equivalent if there exists $ U \in GL_{k|l}( \mathbb{C}) $ such that $ B = U^{-1} A $, where the matrix $ U^{-1} A $ is of the form
\begin{equation}\label{redsuper-basis}
U^{-1} A  =
 \left( \begin{array}{cccc} t & 1 & 0 & \theta \\ \upsilon  & 0 & 1 & u \end{array} \right),
\end{equation}
with  $ t \in \mathbb{C}^k \otimes \mathbb{C}^{n - k} $, $ \theta \in \mathbb{C}^k \otimes \mathbb{C}^{m - l} $, $ \upsilon \in \mathbb{C}^l \otimes \mathbb{C}^{n - k} $ and $ u \in \mathbb{C}^l \otimes \mathbb{C}^{m - l} $. As $ t, u, \theta, \upsilon $ are arbitrary, we obtain a map $ \varphi: G_{k|l , n|m} \to \mathbb{C}^{(n - k)k|(m - l)l} $. As there are
 $ { n \choose k } { m \choose l } $ such maps, we can cover $ G_{k|l , n|m} $ with precisely $ { n \choose k } { m \choose l }$ super-charts.

To finish the construction of super-atlas on $ G_{k|l , n|m} $, we have to define the gluing condition on the intersections. Let $ \Omega $ be the chart associated to (\ref{redsuper-basis}) and let $ \Omega' $ be the chart associated to
\begin{equation}\label{}
(U')^{-1} A  =
 B'  =
 \left( \begin{array}{cccc} t' & 1 & 0 & \theta' \\ \upsilon' & 0 & 1 & u' \end{array} \right).
\end{equation}
Then the relation between the maps is given by
$ B' =  (U')^{-1} A = (U')^{-1} U B $, corresponding to the gluing map $ \varphi' \circ \varphi^{-1}\in \Aut_{\mathbb{C}}(\mathbb{C}^{(n - k)k|(m - l)l})$.

\begin{exm}[The super-Grassmannian $ G_{2|2 , 4|4} $]
Let
\begin{equation}\label{}
A  =
 \left( \begin{array}{cccccccc} x_{11} & x_{12} & x_{13} & x_{14} & \xi_{11} & \xi_{12} & \xi_{13} & \xi_{14} \\
x_{21} & x_{22} & x_{23} & x_{24} & \xi_{21} & \xi_{22} & \xi_{23} & \xi_{24} \\ \eta_{11} & \eta_{12} & \eta_{13} & \eta_{14} & y_{11} & y_{12} & y_{13} & y_{14} \\ \eta_{21} & \eta_{22} & \eta_{23} & \eta_{24} & y_{21} & y_{22} & y_{23} & y_{24} \end{array} \right).
\end{equation}
Then $ U^{-1} A = B $ is of the form
\begin{equation}\label{}
B   =
 \left( \begin{array}{cccccccc} t_{11} & t_{12} & 1 & 0 & 0 & 0 & \theta_{13} & \theta_{14} \\
t_{21} & t_{22} & 0 & 1 & 0 & 0 & \theta_{23} & \theta_{24} \\ \upsilon_{11} & \upsilon_{12} & 0 & 0 & 1 & 0 & u_{13} & u_{14} \\ \upsilon_{21} & \upsilon_{22} & 0 & 0 & 0 & 1 & u_{23} & u_{24} \end{array} \right),
\end{equation}
and $ (U')^{-1} A = B' $ is of the form
\begin{equation}\label{}
B'  =
 \left( \begin{array}{cccccccc} t_{11} & 1 & t_{13} & 0 & 0 & 0 & \theta_{13} & \theta_{14} \\
t_{21} & 0 & t_{23} & 1 & 0 & 0 & \theta_{23} & \theta_{24} \\ \upsilon_{11} & 0 & \upsilon_{13} & 0 & 1 & 0 & u_{13} & u_{14} \\ \upsilon_{21} & 0 & \upsilon_{23} & 0 & 0 & 1 & u_{23} & u_{24} \end{array} \right).
\end{equation}
The matrix $ U $ is formed from $ A $ by choosing columns $3$, $4$, $5$, and $ 6 $ and $ U' $ is formed from $ A $ by choosing columns $2$, $4$,
$5$, and $ 6 $. They are, respectively, of the form
\begin{equation}\label{}
 \left(
 \begin{array}{cccc}
 x_{13} & x_{14} & \xi_{11} & \xi_{12} \\
 x_{23} & x_{24} & \xi_{21} & \xi_{22} \\
 \eta_{13} & \eta_{14} & y_{11} & y_{12} \\
 \eta_{23} & \eta_{24} & y_{21} & y_{22}
 \end{array}
 \right)
\hspace{4mm}
\mbox{ and }
\hspace{4mm}
 \left(
 \begin{array}{cccc}
 x_{12} & x_{14} & \xi_{11} & \xi_{12} \\
 x_{22} & x_{24} & \xi_{21} & \xi_{22} \\
 \eta_{12} & \eta_{14} & y_{11} & y_{12} \\
 \eta_{22} & \eta_{24} & y_{21} & y_{22}
 \end{array}
 \right).
\end{equation}

On the intersection of the maps
$\Omega $ and $ \Omega ' $,
both $ U $ and $ U' $ are invertible. Thus $ U^{-1} $ and $ (U')^{-1} $
 are well-defined. This implies that the product $ (U')^{-1} U $ is also well-defined,
 providing a transition map between the charts corresponding to $ B $ and $ B'$, which is $(U')^{*} U$.\footnote{It does not matter whether we take a general or the unitary point of view, but the computations are simpler in the unitary setting.}
\end{exm}

\subsection{Change of super-variables}\label{subsection:change-of-variables}

Having the odd part of the integral defined, we introduce the full integral of a (compactly supported) function on the superspace $ \mathbb{C}^{n|m} $. The function $ \varphi : \mathbb{C}^{n|m} \to \mathbb{C} $ can be expanded into a power series in the odd variables as
\begin{equation}\label{}
\varphi (x, \xi) \, = \,
 \varphi_0 (x) + \sum_{i = 1}^m \varphi_1 (x) \xi_i
 + \ldots + \varphi_{1, 2, \ldots , m} (x) \xi_1 \xi_2 \cdots \xi_m , \nonumber
\end{equation}
where, this time, each coefficient $\varphi_{\alpha}(x)$ is an element of the set $ C_c^{\infty} (\mathbb{C}^n)$ of compactly-supported smooth functions on $\mathbb{C}^n$. This implies that the integral $ \int \varphi \: dx  d\xi $ is of the form
\begin{equation}\label{eqn:power-series-odd-variables}
\int_{\mathbb{C}^{n|m}} \varphi (x, \xi) dx d\xi  =
 \int_{\mathbb{C}^n} \varphi_{1, 2, \ldots , m} (x) \, dx
\end{equation}
by using properties of Berezin integral, i.e., $ \int_{\mathbb{C}^{0|1}} d\xi = 0 $ and $ \int_{\mathbb{C}^{0|1}} \xi d\xi = 1 $.
We thus have a super-analogue of Fubini's theorem:
\begin{equation}
\int_{\mathbb{C}^{n|m}}  \varphi(x,\xi) dx d\xi =
\int_{\mathbb{C}^n} \varphi_{1, 2, \ldots , m} (x) dx \int_{\mathbb{C}^{0|m}}\xi_1\xi_2\cdots \xi_m d\xi.
\end{equation}

Now, the following analysis is an important generalization of classical concepts in analysis, geometry and topology. To proceed, we first deliver a formula for the change of super-variables, paving way to prove the inverse and implicit function theorems, leading us to an analysis on supermanifolds.

Let $ (y, \eta) = f (x, \xi) $. The function $ f $ has the derivative of the form\footnote{The partial derivatives $ \partial y / \partial x $, $\partial y/\partial \xi$, $\partial \eta/\partial x$, and $\partial \eta/\partial \xi$ denote the blocks of super-derivative.}
\begin{equation}\label{}
D f  =
 \left( \begin{array}{cc} \dfrac{\partial y}{\partial x} & \dfrac{\partial y}{\partial \xi} \\ \dfrac{\partial \eta}{\partial x} & \dfrac{\partial \eta}{\partial \xi} \end{array} \right).
\end{equation}
The Jacobian, however, cannot be computed in the usual way. Instead, one uses the super-determinant, or {\em Berezinian}, defined as
\begin{equation}\label{}
\mathrm{sdet} \, D f  =
 \det \left( \dfrac{\partial y}{\partial x} - \dfrac{\partial y}{\partial \xi} \left( \dfrac{\partial \eta}{\partial \xi}\right)^{-1} \dfrac{\partial \eta}{\partial x} \right) \det \left( \dfrac{\partial \eta}{\partial \xi}\right)^{-1} , \nonumber
\end{equation}
where $ \partial \eta / \partial \xi $ must be invertible.
Thus we have the change of super-variables, and the corresponding integrals
\begin{equation}\label{}
\int \varphi (y, \eta) \: d y d \eta
\end{equation}
and
\begin{equation}\label{}
\int \varphi ( f (x, \xi))
\det \left( \dfrac{\partial y}{\partial x} - \dfrac{\partial y}{\partial \xi} \left( \dfrac{\partial \eta}{\partial \xi}\right)^{-1} \dfrac{\partial \eta}{\partial x} \right) \det \left( \dfrac{\partial \eta}{\partial \xi}\right)^{-1}
d x d \xi \nonumber
\end{equation}
are equal.\footnote{We do not need to insert an absolute value around the two determinants in the integral since the Berezinian is an even function.}

\begin{rem}
The change of super-variables is implicitly and minimally used in the construction of beta super-integral (see the proof of Theorem~\ref{thm:Euler-super-integral} in Section~\ref{subsection:special-case}).
\end{rem}

\begin{comment}
Another construction is via the super-unitary groups. Let $ U (n|m) $. The super-unitary basis of a subspace $ V_{k|l} $ can be described as the
 super-unitary basis of $ \mathbb{C}^{n|m} $ modulo the action of $ U (n - k|m - l) $, which sets the $ n + m - k - l $ vectors, leaving the remaining $ k + l $ vectors arbitrary.

\begin{defi}[Stiefel super-manifold]
The homogenous space $ S_{k|l , n|m} : = U (n|m) / U (n - k|m - l) $ is called \emph{Stiefel super-manifold}.
\end{defi}

In this super-unitary seting, the action of $ U (k|l) $ on the set of unitary bases of $ V^{k|l} $, i.e. on $ S_{k|l, n|m} $, corresponds to the action of $ GL (k|l , \mathbb{C}) $ on $ B $. Thus the definition of $ G_{k|l , n|m} $ is euivalent to the following one.

\begin{defi}[Super-Grassmannian]
The homogenous space $ G_{k|l , n|m} : = U (n|m) / ( U (k|l) \times U (n - k|m - l)) $ is called the \emph{Grassmannian super-manifold}.
\end{defi}

\begin{exr}
Proove that the both Definitions of $ G_{k|l , n|m} $ are indeed equivalent.
\end{exr}
\end{comment}

\section{Beta super-integral}\label{section:classical-super-beta-integral}

For $l\leq m$,
let $ A' $ be the super-matrix of the form
\begin{equation}
\left(
\begin{matrix}
x' & \xi' \\
\eta' & y'
\end{matrix}
 \right),
\end{equation}
with $ x' \in \mathbb{C}^2 \otimes \mathbb{C}^3 $, $ \xi' \in \mathbb{C}^2 \otimes \mathbb{C}^m $, $ \eta' \in \mathbb{C}^l \otimes \mathbb{C}^3 $ and $ y' \in \mathbb{C}^l \otimes \mathbb{C}^m $ such that
$ A' $ has maximal rank.
Then we can regard $ A' $ as a supersymmetric extension of a $ 2 \times 3 $ matrix $ x'$, of maximal rank. We can also identify $A'$ with the set of $2+l$ vectors in the vector superspace $ \mathbb{C}^{3|m}$.
 Because of this, one can regard $ A' $ as a point of the Grassmannian manifold $ G_{2|l, 3|m} $. Assuming that the invertible $ 2|l \times 2|l $ minor $ U $ is constructed from the columns
$2$, $3$, $4, \ldots$, $l + 3 $ (see Section~\ref{subsection:properties-super-Grass} for more detail on our choice for these columns),
we can construct a map\footnote{This map is actually a super-biholomorphic isomorphism $ \varphi : G_{k|l , n|m} \to \mathbb{C}^{k|l \times  (n - k|m - l)} $ since $G_{k|l , n|m}$ is locally isomorphic to $\mathbb{C}^{k|l \times  (n - k|m - l)}$.
But since  $ \mathbb{C}^{k|l \times  (n - k|m - l)} $ and $ \mathrm{Hom} (\mathbb{C}^{k|l}, \mathbb{C}^{n - k | m - l}) $ are the same as vector superspaces, we will identify the two.}
 $ A = U^{-1} A' $, which is of the form
\begin{equation}\label{eqn:matrixA-G-23-LM}
A  =
 \left( \begin{matrix} x & I_2 & 0 & \xi \\ \eta & 0 & I_l & y
\end{matrix}
 \right) ,
\end{equation}
with $ x \in \mathbb{C}^2 \otimes \mathbb{C} $, $ \xi \in \mathbb{C}^2 \otimes \mathbb{C}^{m - l} $, $ \eta \in \mathbb{C}^l \otimes \mathbb{C} $ and $ y \in \mathbb{C}^l \otimes \mathbb{C}^{m - l} $,
with all nonconstant entries being arbitrary.

To $ x $ one can (in the classical setting) associate the function
\begin{equation}
\Phi (s, x) \, := \,
 \int_{\gamma_x} \omega_x ,
\end{equation}
where $ \omega_x =
 l_1^{s_0} l_2^{s_1} l_3^{s_2} \, dt $,
 $l_1 = x_{11}+x_{21}t$, $l_2 = 1$, $l_3 = t$,
 and
 $ \gamma_x $ is a cycle associated to a $ 3 $-point arrangement in $ \mathbb{CP}^1 $, bounded by the equations $  l_i = 0 $.

In what follows, we will describe the super-generalization of this construction.

\subsection{Construction of the general beta super-integral}\label{subsection:general-beta-integral}

 The projective superspace $\mathbb{CP}^{1|l}$ has homogeneous coordinates $[ t'_1 : t'_2
 | \theta'_1 , \ldots , \theta'_l ] $.
  If $t_1'\not=0$, then $[ t'_1 : t'_2
 | \theta'_1 , \ldots , \theta'_l ] = [1 : t
 | \theta_1 , \ldots , \theta_l ]$, which can be viewed as
the point
\begin{equation}
( 1 : t
 | \theta_1 , \ldots , \theta_l ) \in \mathbb{C}^{1|l} .
\end{equation}

The reduced matrix $ A $ in \eqref{eqn:matrixA-G-23-LM} acts on $ \mathbb{CP}^{1|l} $ from the right as multiplication (cf. \cite{MR3588978}):
\begin{equation}\label{eqn:matrixA-G-23-LM-multn}
[ t'_1 : t'_2
 | \theta'_1 , \ldots , \theta'_l ]
 \left( \begin{matrix} x & I_2 & 0 & \xi \\ \eta & 0 & I_l & y
\end{matrix}
 \right).
\end{equation}
Thus it also acts on the space $ \mathbb{C}^{1|l} $ from the right, and the result is a family of affine forms: three even\footnote{By even, we mean that the variables
$ \xi : = \eta_{ij} \theta_i $ and $ \xi' : = \eta_{ij}' \theta_i' $ satisfy
$ \xi ' \xi = \eta_{ij}' \theta_i' \eta_{ij} \theta_i = \eta_{ij} \eta_{ij}' \theta_i' \theta_i = \eta_{ij} \theta_i \eta_{ij}' \theta_i' = \xi \xi' $, so the minus sign does not apply in their case when it comes to the commutation relations. Thus they are even.}
  forms:
\begin{align}
l_1 &  =  x_{11} + x_{21} t + \eta_{11} \theta_1 + \eta_{21} \theta_2+ \ldots + \eta_{l1} \theta_l,   \nonumber \\
l_2 &  =  1,
\nonumber \\
l_3 &  = t,

 \nonumber
\end{align}
and $ m $   odd forms:
\begin{align}
\lambda_i & =   \theta_i  \hspace{4mm}\mbox{ for } 1 \leq  i \leq l  ,   \nonumber \\
 \lambda_{l + 1} &  =
 \xi_{1, l + 1} + \xi_{2, l + 1} t + y_{1, l + 1} \theta_1 + \ldots + y_{l, l + 1} \theta_l ,  \nonumber \\
  \lambda_{l + 2} &  =
 \xi_{1, l + 2} + \xi_{2, l + 2} t + y_{1, l + 2} \theta_1 + \ldots + y_{l, l + 2} \theta_l,  \nonumber \\
 &\hspace{2mm} \vdots \, \nonumber \\
 \lambda_m &  =
 \xi_{1, m} + \xi_{2, m} t + y_{1, m} \theta_1 + \ldots + y_{l, m} \theta_l .\nonumber
\end{align}

 \begin{defi}\label{defn:super-beta-integral-general}
The {\em super-beta integral} is defined as
\begin{equation}\label{sup-bet}
\Phi (s|\sigma ; x, y, \xi, \eta) \, : = \,
  \int_{\gamma_x \times \mathbb{C}^{0|l}} l_1^{s_1} l_2^{s_2} l_3^{s_3} \lambda_1^{\sigma_1} \cdots \lambda_m^{\sigma_m} dt \wedge d \theta_1 \wedge \cdots \wedge d \theta_l .
\end{equation}
\end{defi}

\subsection{Special case of the beta super-integral}\label{subsection:special-case}
We will first prove Theorem~\ref{thm:Euler-super-integral}.

\begin{proof}
Consider the matrix
\begin{equation}
A  =
 \left(
 \begin{array}{ccccc}
 x_{11} & x_{12} & x_{13} & \xi_{11} & \xi_{12} \\
 x_{21} & x_{22} & x_{23} & \xi_{21} & \xi_{22} \\
 \eta_{11} & \eta_{12} & \eta_{13} & y_{11} & y_{12}
 \end{array} \right),
\end{equation}
of maximal rank. This matrix can be identified with the set of $ 3 $ vectors in the vector superspace $ \mathbb{C}^{3|2} $. Furthermore, as the rank is maximal, the image of the map that corresponds to $ A $ is a $ 2 | 1 $-dimensional super-subspace since the rows correspond to linearly independent vectors, i.e., $2$ even vectors and $1$ odd vector.

As $ A $ is of maximal rank, there exists a non-zero minor, say $ U $. Without loss of generality, let's assume that $ U $ is of the form
\begin{equation}
U  =
 \left(
 \begin{array}{ccc}
 x_{12} & x_{13} & \xi_{11} \\
 x_{22} & x_{23} & \xi_{21} \\
 \eta_{12} & \eta_{13} & y_{11}
 \end{array} \right).
\end{equation}
Then the $ 2|1 $-subspace of $ \mathbb{C}^{3|2} $ that is described by $ A $ is also described by  $U^{-1}A$, which is of the form
\begin{equation}
U^{-1} A   =
 \left(
 \begin{array}{ccccc}
 x'_{11} & 1 & 0 & 0 & \xi'_{12} \\
 x'_{21} & 0 & 1 & 0 & \xi'_{22} \\
 \eta'_{11} & 0 & 0 & 1 & y'_{12}
 \end{array} \right).
\end{equation}
With the data provided by $ U^{-1}A $, we can form a family of two linear forms, $ x'_{11} t_1 + x'_{21} t_2 + \eta'_{11} \theta_1 $ and $ \xi'_{12} t_1 + \xi'_{22} t_2 + y_{12}' \theta_1 $, together with three characters: $2$ even ones $ t_1, t_2 $ and $1$ odd one $ \theta_1 $, by multiplying on the left by $(t_1,t_2;\theta_1)$.

The affine counterparts are
\begin{equation}\label{eqn:affine-counterparts}
\begin{split}
l_1 & =  1 + x_{21}' t + \eta_{11}' \theta,  \\
l_2 & =  \xi_{12}' + \xi_{22}' t + \theta ,  \\
\end{split}
\end{equation}
where $x_{11}'$ and $y_{12}'$ in \eqref{eqn:affine-counterparts} are canceled under the change of variables.

For the ease of reading, let us omit the primes over the coefficients.
The group, which depends locally,
acts on the set of linear forms $l_i$, which, in fact, acts on the powers $l_i^{\alpha_i}$ of the linear forms.
After fixing exponents, one constructs the form
\begin{equation}
\omega \, := \,
 t^s l_1^{p_1} l_2^{p_2} \theta \: d \theta \: d t ,
\end{equation}
which can be integrated (for certain values of $ s, p_1, p_2 $) over a region $ \gamma $ in $ \mathbb{C}^{1|1}=\Spec(\mathbb{C}[t;\theta])$.
 The first factor constrains us to $ t > 0 $, while from $ l_1 $ it follows that
 $ t > - x_{21}^{-1} $
since if in, say, $ (1 - tx)^{\alpha} $, we allow both $ t x < 0 $ and $ t x > 0 $,
then there is a jump coming from the multi-valuedness of $ (1 - tx)^{\alpha} $.\footnote{In the classical case of logarithms, choosing a loop around the singularity and running around brings one to another point other than the original point (the new point will differ by $\pm 2 \pi $, depending on the orientation of the loop).}
This can be overcome by considering the universal covering for complex spaces and their monodromy, or, so called, twisted cohomology (but in this manuscript, we will rule out these cases by restricting to the parameters of the classical beta function).

The parameter $ \theta $ is allowed to be arbitrary. Thus, we get the following formula:
\begin{equation}
\int_{\gamma} \omega \, := \,
 \int_{- 1/x_{21}}^0 \int_{-\infty}^{\infty}
 t^s (1 + x_{21} t + \eta_{11} \theta)^{p_1} (\xi_{12} + \xi_{22} t + \theta)^{p_2} \theta \: d \theta d t .
\end{equation}
This can be further simplified using the change of variables
$ x_{21} t = - u $, with $ dt = - du / x_{21} $:
\begin{equation}
\int_{\gamma} \omega  =
- (- x_{21})^{-s - 1} \int_0^1 \int_{-\infty}^{\infty} u^s (1 - u + \eta_{11} \theta)^{p_1} (\xi_{12} - \xi_{22} x_{21}^{-1} u + \theta)^{p_2} \theta \: d \theta d u .
\end{equation}
Further simplification is done by letting $\eta = \eta_{11}$, $\xi = \xi_{12}$ and
$\xi'= - \xi_{22} x_{21}^{-1} $ as only the integral is of importance.
This concludes the proof.
\end{proof}

\begin{rem}\label{rem:Theorem1-monodromy}
We will discuss in our future work the case when $t x_{21}+1 \leq 0$ for Theorem~\ref{thm:Euler-super-integral} by considering monodromy for the super-variables.
\end{rem}

We will now prove Theorem~\ref{thm:Euler-super-integral-relation}.
\begin{proof}
By Theorem~\ref{thm:Euler-super-integral}, we have
\begin{equation}
\Phi(s,p_1,p_2; \xi, \xi', \eta)  = - (-x_{21})^{-s-1} \int_0^1 \int_{-\infty}^{\infty} u^s (1 - u + \eta \theta)^{p_1} (\xi + \xi' u + \theta)^{p_2} \theta \: d \theta d u .
\end{equation}
Since $B(s,p_1,p_2; \xi, \xi', \eta)$ differs from $\Phi(s,p_1,p_2; \xi, \xi', \eta)$ by a scalar,
we conclude that
$$
\Phi(s,p_1,p_2; \xi, \xi', \eta)  = - (-x_{21})^{-s-1} B(s,p_1,p_2; \xi, \xi', \eta),
$$
where
\[
B(s,p_1,p_2; \xi, \xi', \eta) = \int_0^1 \int_{-\infty}^{\infty} u^s (1 - u + \eta \theta)^{p_1} (\xi + \xi' u + \theta)^{p_2} \theta \: d \theta d u .
\]
\end{proof}

\section{Integrating the super-beta integral with respect to the odd variables}\label{section:super-beta-integral-odd-variables}

We will finally prove Theorem~\ref{thrm:super-beta-gamma-function}.

\begin{proof}
Recall the binomial expansion
\begin{equation}
(x + y)^p \, = \,
 \sum_{k \geq 0} { p \choose k } x^{p - k} y^k .
\end{equation}
Applying it to powers of the affine forms under the integral in Theorem~\ref{thm:Euler-super-integral},
 we get
\begin{equation}\label{eqn:affine-forms-power-series-expansion}
\begin{split}
(1 - u + \eta \theta)^{p_1} & =
 \sum_{k \geq 0} { p_1 \choose k } (1 - u)^{p_1 - k} (\eta \theta)^k,  \\
(\xi + \xi' u + \theta)^{p_2} & =
 \sum_{k \geq 0} { p_2 \choose k } (\xi + \xi' u)^{p_2 - k} \theta^k , \\
 \end{split}
\end{equation}
but $ \theta^2 = 0 $ and $ \eta^2 = 0 $
by supersymmetry. That is,
$ \xi_i \xi_j = - \xi_j \xi_i $ for any super-variable $ \xi_i $,
so $ \xi_i^2 = - \xi_i^2 $.
This implies that $ \xi_i^2 = 0 $.
It follows that $ (\eta \theta)^2 = 0 $.
The equalities in \eqref{eqn:affine-forms-power-series-expansion} simplify to
\begin{equation}\label{eqn:eqn:affine-forms-poly}
\begin{split}
(1 - u + \eta \theta)^{p_1} & =
 (1 - u)^{p_1} + p_1 (1 - u)^{p_1 - 1} \eta \theta ,  \\
(\xi + \xi' u + \theta)^{p_2} & =
 (\xi + \xi' u)^{p_2} + p_2 (\xi + \xi' u)^{p_2 - 1} \theta . \\
 \end{split}
\end{equation}
Multiplying the two equalities in \eqref{eqn:eqn:affine-forms-poly} together, we get\footnote{Variables $ \xi $ and $ \xi' $ are odd; however, the parity operator is not relevant as the only summand that contributes to the integral is the one with $ 0 $-th power of $ \theta $.}
\begin{equation}\label{eqn:multi-two-eqns}
\begin{split}
(1 - u + \eta &\theta)^{p_1} (\xi + \xi' u + \theta)^{p_2}
=
 (1 - u)^{p_1} (\xi + \xi' u)^{p_2}  \\
 &\hspace{4mm}
 +  \cancel{p_1 (1 - u)^{p_1 - 1} \eta \theta  (\xi + \xi' u)^{p_2}}
 +  \cancel{(1 - u)^{p_1}  p_2 (\xi + \xi' u)^{p_2 - 1} \theta }
 + \ldots
 \end{split}
\end{equation}
where the two terms cancel due to supersymmetry.
Since $\int_{-\infty}^{\infty} \theta \, d\theta =1$,
the beta super-integral for $ B $ then reduces to
\begin{equation}\label{eqn:integral-formula-for-B}
\begin{split}
B (s, p_1, p_2; \xi, \xi', \eta)  &=
 \int_0^1 u^s (1 - u)^{p_1} (\xi + \xi' u)^{p_2} \, d u  \\
 &=
 \xi^{p_2} \int_0^1 u^s (1 - u)^{p_1} \left(1-\left( - \frac{\xi'}{\xi}\right) u\right)^{p_2} \, du.
 \end{split}
\end{equation}
Since
\[
\beta(s+1,p_1+1) = \frac{\Gamma(s+1)\Gamma(p_1+1)}{\Gamma(p_1+s+2)}
\hspace{4mm}
\mbox{ for } s+1,p_1+1 >0
\]
(see
Theorem 7 on page 19 in \cite{MR0107725} or
Theorem 2.5.9 on page 46 in \cite{hannah2013identities}),
and since Euler-Gauss hypergeometric function (see Theorem 3.4.1 on page 71 in \cite{hannah2013identities}) gives us
\[
_2F_1 \left( \begin{matrix} -p_2 , s+1 \\ p_1 + s+2 \end{matrix} \, \Big| - \frac{\xi'}{\xi}  \right)
=  \frac{\Gamma(p_1+s+2)}{\Gamma(s+1)\Gamma(p_1+1)}
\int_0^1 u^s (1-u)^{p_1}  \left(1-\left( - \frac{\xi'}{\xi}\right) u\right)^{p_2} \, du,
\]
we have
\[
\xi^{p_2}  \,
{}_2F_1 \left( \begin{matrix} -p_2 , s+1 \\ p_1 + s+2 \end{matrix} \, \Big| - \frac{\xi'}{\xi}  \right)
=
\frac{\Gamma(p_1+s+2)}{\Gamma(s+1)\Gamma(p_1+1)} B (s, p_1, p_2; \xi, \xi', \eta).
\]
We conclude
\[
B (s, p_1, p_2; \xi, \xi', \eta)
=
\frac{\Gamma(s+1)\Gamma(p_1+1)}{\Gamma(p_1+s+2)}
\xi^{p_2}  \,
{}_2F_1 \left( \begin{matrix} -p_2 , s+1 \\ p_1 + s+2 \end{matrix} \, \Big| - \frac{\xi'}{\xi}  \right).
\]
\end{proof}

\section*{Appendix}

We will now state and prove Ramanujan's master theorem for completeness. Also see \cite{MR2994092}, after Theorem 3.2 on page 4.

\begin{thrm}\label{thm:sinphi}
If
\begin{equation}
\frac{\pi}{\sin \pi s} F (- s) \, = \,
 \int_0^{\infty} t^{s - 1} \varphi (t) \, dt ,
\end{equation}
then
\begin{equation}
\varphi (t) \, = \,
 \sum_{n \geq 0} (- t)^n F (n).
\end{equation}
\end{thrm}

\begin{proof}
If
\begin{equation}
F (s) \, = \,
 \int_0^{\infty} t^{s - 1} f (t) \, dt
 \quad
 \mbox{ and }
 \quad
G (s) \, = \,
 \int_0^{\infty} t^{s - 1} g (t) \, dt , \nonumber
\end{equation}
then by Mellin convolution, we have
\begin{equation}
F (s) G (s) \, = \,
 \int_0^{\infty} t^{s - 1} h (t) \, dt ,
\end{equation}
where
\begin{equation}
h (t) \, = \,
 \int_0^{\infty} \frac{f (u) g (t / u)}{u} \, du .
\end{equation}

In particular,
\begin{equation}\label{eqn:Ramanujan-proof-end}
\frac{\pi}{\sin \pi s} F (- s) \, = \,
 \int_0^{\infty} t^{s - 1} \varphi (t) \, dt ,  \quad \mbox{ where }
 \varphi (t) \, = \,
 \int_0^{\infty} \frac{f (1/u)}{u + t} \, du .
\end{equation}
We have $ F (- s) $ in place of $ F (s) $, which translates from $  f (u) $ to $ f (1/u) $:
\[
F(-s) = \int_0^{\infty} u^{s-1} f(1/u)\, du.
\]
Also,
$$ G (s) = \frac{\pi}{\sin \pi s} = \int_0^{\infty} \frac{t^{s - 1}}{1 + t}\, dt ,
$$
so the convolution of $F(-s)$ with $ \pi / \sin \pi s $ is
\begin{equation}\label{}
\frac{\pi}{\sin \pi s} F (- s) \, = \,
 \int_0^{\infty} t^{s - 1} \int_0^{\infty} \frac{f (1/u)}{1 + t/u} \, \frac{du}{u} \, dt = \,
 \int_0^{\infty} t^{s - 1} \int_0^{\infty} \frac{f (1/u)}{u + t} \, du
 \, dt.
\end{equation}

Replacing $ (u + t)^{-1} $ by its Taylor series, we obtain
\begin{align*}
 \int_0^{\infty} \frac{f (1/u)}{u + t} \, du
 &=  \int_0^{\infty}  \frac{u^{-1} f (1/u)}{1 + (t/u)} \, du \\
 &=  \int_0^{\infty} \frac{f(1/u)}{u} \sum_{n\geq 0} (-t/u)^n du \\
 &= \sum_{n\geq 0}  (-t)^n \int_0^{\infty} u^{-n-1} f(1/u)\, du.
\end{align*}
Now, letting $v=1/u$ (and so $dv = -u^{-2}du$),  we see that
\begin{align*}
\int_0^{\infty} u^{-n-1} f(1/u)\, du &= \int_0^{\infty} v^{n+1}f(v)\frac{dv}{v^2}
= \int_0^{\infty} v^{n-1}f(v)\, dv
= F(n).
\end{align*}
\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
Graduate School of Theology
Meet the Department
Wendy Farley, Ph.D.
Director, Program in Christian Spirituality, Rice Family Professor of Spirituality
PhD, Vanderbilt University
MA, Vanderbilt University
Montgomery Hall, Marin Campus
E: wendy_farley@redlands.edu
E: wfarley@sfts.edu
Meet Wendy
A leading theologian, Wendy Farley has written extensively on women theologians and mystics, religious dialogue, classical texts, contemporary ethical issues, and contemplative practices.
Professor Farley received her Ph.D. from Vanderbilt University in 1988. Her teaching and research interests include women theologians, Buddhist-Christian dialogue, spirituality and social justice, classical texts, and contemplative practices. Her first book, Tragic Vision and Divine Compassion: A Contemporary Theodicy (Westminster John Knox, 1990) considers the problem of evil by focusing on suffering rather than sin and abandons the forensic model of God in favor of one emphasizing compassion as a dominant metaphor for the divine. A second work, Eros for the Other: Retaining Truth in a Pluralistic World (Penn State: 1996), also takes up the relationships between ethical and philosophical issues in religion. In 2005, she published The Wounding and Healing of Desire: Weaving Heaven and Earth (Westminster John Knox), which combines attention to contemplative practices, folk traditions, and inter-religious dialogue to reflect on suffering and transformation. Gathering Those Driven Away: a Theology of Incarnation (Westminster John Knox, 2011), reflects on the meaning of Christian faith and tradition for women, queers, and others that the church has had difficulty recognizing as part of the body of Christ. She also recently edited (with Emily Holmes) a collection of essays called Women, Writing, Theology: Transforming a Tradition of Exclusion.
Her latest book, "The Thirst of God: Contemplating God's Love with Three Women Mystics" (Westminster John Knox, 2015), explores the spirituality of medieval mystics Marguerite Porete, Mechthild of Magdeburg, and Julian of Norwich. In Farley's words, "These women have important things to tell us about our faith, the same as contemporary contemplatives, with the emphasis on divine love."
Orientation to Theological Education (SFTS)
Systematic Theology (SFTS)
Compassion and Contemplation (SFTS)
Contemplating Beauty (Emory University)
Contemplative Theology: Plato to Porete (Emory University)
Women Mystics and Theologians (Emory University)
Religion and Social Change (Emory University)
The Thirst of God: Contemplating God's Love with Three Women Mystics, Westminster/John Knox, 2015 (169 pp)
Gathering Those Driven Away: a Theology of Incarnation, Westminster/John Knox, 2011 (233 pp)
The Wounding and Healing of Desire: Weaving Heaven and Earth, Westminster/John Knox, 2005 (180 pp)
Eros for the Other: Retaining Truth in a Pluralistic Age, Pennsylvania State Press, 1996 (220pp)
Tragic Vision and Divine Compassion: A Contemporary Theodicy, Westminster/John Knox, Spring, 1990 (150 pp)
EDITED VOLUME
Women Writing Theology: Transforming a Tradition of Exclusion, co-editor with Emily Holmes, Waco, TX: Baylor University Press, 2011 (330pp)

redpajama_set_name: RedPajamaCommonCrawl


text:
Karl Ritter (* 22. oder 23. Dezember 1959 in Stockerau, Niederösterreich) ist ein österreichischer Musiker und Komponist.

Biographie
Mit sechs Jahren lernte Ritter Geige, wechselte aber im Alter von 12 Jahren zur Gitarre und begann sehr bald mit Improvisation und Komposition eigener Stücke.

1980 produzierte Ritter seine erste Langspielplatte, "Ex-podo-Z". Es folgten ab 1985 Auftragsarbeiten für das Wiener Renaissance Theater sowie Hörspielproduktionen für den ORF und den Bayerischen Rundfunk.

Größere Bekanntheit erlangte Ritter unter dem Namen "Leopold Karasek" als Mitglied der Gruppe um Willi Resetarits alias Kurt Ostbahn, der er von 1988 bis 1994 ("Ostbahn-Kurti & die Chefpartie") und von 1996 bis 2003 ("Kurt Ostbahn & die Kombo") angehörte.

Ritter spielte mehrere LPs und CDs als Solokünstler sowie als Mitglied verschiedener Formationen ein. Er wirkte auch bei Projekten – vor allem österreichischer Künstler – aus Musik, Kabarett und darstellender Kunst mit. Darüber hinaus schuf er Kompositionen für Theater, Film und Fernsehen.

Veröffentlichungen

Alben
 1994: Perikato - Blackbox I
 1995: Karl Ritter - Dobromann
 1996: Ritter Pernes Eder - Sonnwende
 1996: Karl Ritter - Schwarzfahrer
 1998: Sel Gapu Mex - Sturm
 1999 Karl Ritter-Stick to it
 2000: Perikato – Blackbox II
 2002: Ritter Lechner Coleman – Nuages
 2002: Sel Gapu Mex – Verliebt
 2002: Karl Ritter-Schattenklänge
 2002: Takeya/Ritter/Pirker/Tang-Rainbow Session
 2003: Windhund – Mwapona Windhund
 2003: Windhund – Zimbabwe Rewind
 2004: Karl Ritter – Atmen
 2005: Windhund – Soundbridge / Klangbrücke
 2005: Karl Ritter – Live (Atmen, Dobromann, Schattenklänge)
 2007: Karl Ritter – Blau
 2007: Karl Ritter – Rot
 2007: Weisse Waende – Nur für kurze Zeit
 2009: Karl Ritter – Traumland
 2009: Weisse Waende – Frei erfunden
 2011: Karl Ritter-Soundritual
 2014: Karl Ritter-Soundkur
 2015: Weisse Waende-33-44-55
 2017: Weisse Waende-10
 2020: Karl Ritter - Soundritual live
 2020: Karl Ritter - Crashtestjazz 1
 2020: Karl Ritter - daham gspüt, unheimelige Aufnahmen 2000–2020
 2021: Karl Ritter - KOMBOjaner

DVD
 1999: Roland Düringer: Benzinbrüder/Die Show
 2003: Karl Ritter – Waunsinn...
 2006: Karl Ritter-Live: Atmen, Dobromann, Schattenklänge
 2010: Karl Ritter: Am Anfang war die Improvisation
 2020: Karl Ritter-Guitar Driver

Mitwirkung
 1986: Schmetterlinge: Mit dem Kopf durch die Wende
 1988: Ostbahn Kurti & die Chefpartie: Liagn und lochn
 1991: Ostbahn Kurti & die Chefpartie: 1/2 so wüd
 1992: Ostbahn Kurti & die Chefpartie: a blede Gschicht
 1994: Ostbahn Kurti & die Chefpartie: Saft & Kraft
 1994: Ostbahn Kurti & die Chefpartie: Trost & Rat
 1995: Kurt Ostbahn & die Kombo: Espresse Rosi
 1997: Kurt Ostbahn & die Kombo: Reserviert fia zwa
 1997: Kurt Ostbahn: Blutrausch
 1998: Otto Lechner: Seventh Heaven
 1998: S.T.S.: Volle Kraft
 1999: Kurt Ostbahn & die Kombo: 50 verschenkte Jahre
 2001: Kurt Ostbahn & die Kombo: Ohjo
 2002: Joanna Lewis & Herbert Reisinger: Songs for the Boys
 2003 Otto Lechner: Fleckerlteppich
 2003: Kurt Ostbahn & die Kombo: Wann die Musik...
 2003: Kurt Ostbahn & die Kombo: vorbei is.
 2003: Kurt Ostbahn & die Kombo: Hohe Warte live-1,2,3
 2003: S.T.S.: Herzverbunden
 2004: Kurt Ostbahn trifft Sivan Perwer: live
 2004: Birgit Denk: hoits eich au!
 2005: Kurt Ostbahn & die Kombo: Höchste Zeit
 2005: Kadero Rai: Perdu
 2006: Thomas Pernes: Sonnwende – Alpenglühen
 2007: Dave Gahan: Hourglass
 2008: Wolfgang Mitterer: Sopop
 2010: Otto Lechner & Anne Bennent: Die Stimmen von Marrakesch
 2011: Otto Lechner & Anne Bennent: Leila & Madschnun
 2011: Melissa Coleman: Back Door
 2011: Willi Resetarits: Jesus und seine Hawara
 2012: Philip Scheiner: De Waund

Live-Projekte mit
Musiker (Auswahl)
 Joe Zawinul
 Willi Resetarits (Ostbahn Kurti)
 Otto Lechner
 Joanna Lewis
 Şivan Perwer
 Klaus Trabitsch
 Melissa Coleman
 Franz Hautzinger
 Peter Herbert
 Anne Bennent

Kabarettisten
 Willi Resetarits (Ostbahn Kurti)
 Lukas Resetarits
 Georg Ringsgwandl
 Roland Düringer

Filmmusik
 1996: Schwarzfahrer
 1997: Blutrausch
 2003: Handbikemovie
 2005: Fix Alert
 2012: Killing Time – Zeit zu sterben
 2012: Indian Dreams
 2018: Guitar Driver
 2020: Herbst in Bangkok

Quellen

 Biographie
 Übersicht der Platten, an denen Ritter mitgewirkt hat

Weblinks
 Offizielle Website
 Profil auf Ostbahn.at

Komponist (Österreich)
Filmkomponist
Gitarrist
Österreicher
Geboren 1959
Mann

redpajama_set_name: RedPajamaWikipedia


text:
2020 Prospects
Dodgers Digest Los Angeles Dodgers Baseball Blog
Organizational Depth Chart
2015 Dodgers Top 100 Prospects: Introduction
Dustin Nosler 01/19/2015	2015 Prospects 319 Comments
This is the introductory offering in my 11-part series about the Dodgers' Top 100 prospects, ranked by yours truly. This is how things will break down:
100-76: Short write-ups with best tool, 2015 location and estimated time of arrival in the majors
75-51: Same as 100-76
50-41: At least a solid paragraph, with scouting grades, 2015 location and ETA in the majors
40-31: Same as 50-41
30-21: Full scouting reports/write-ups with scouting grades, 2015 location and ETA in the majors
10-1: Same as 30-21 and 20-11
I will follow-up the Top 100 with supplemental articles in the form of "Best of" the Dodgers' system (tools) and the projected 2018 lineup and pitching staff (2017 projection, for reference).
You can find everything prospects in the aptly named "Prospects" tab on the top of the page. There will be a link to all the 2014 coverage, if you're feeling nostalgic.
I'll be the first to admit that there aren't a ton of guys — especially in the back half of the 100 — who are ever going to sniff the majors. I will also be accused of making up names and ranking players too low, the latter of which is fine. But every player on the list is a real-life human baseball player (except maybe Julio Urias, because he's clearly not of this world).
It's nice to have at least some information on these guys. I mean, Jose De Leon — a player which almost no one knew about before last season, would have been in this 51-100 range of my Top 50 last year, because he, obviously, wasn't in the Top 50.
All players who have not reached 50 innings pitched or 130 at-bats in the MLB, and have less than 45 days of pre-Sept. 1 service time in the MLB are eligible for this list. Ages are the "age" season for the player, i.e. "his age-23 season."
Prospect System Grade
Impact Potential: A
The Dodgers didn't lose any of their top three prospects — Joc Pederson, Corey Seager, Julio Urias — to trades or graduations, added a legitimate mid-rotation starter in Grant Holmes and potential center fielder in Alex Verdugo in the first two rounds of the 2014 MLB Draft. They also had guys like Jharel Cotton, Jose De Leon and Julian Leon make huge jumps in the low minors, which helps the overall bottom line.
Depth: B-
The Dodgers did lose the likes of Jonathan Martinez, Jesmuel Valentin and Victor Arano, but they also gained guys like Austin Barnes, Adam Liberatore, Shawn Zarraga, as well as other 2014 draftees. I graded the depth as a "C+" last year, so it only stands to get better from additions and some guys taking steps forward. Some guys took a step back (Jacob Scavuzzo, Alex Santana), but the depth is as good as it has been in recent years.
Overall: B+
This is the strongest the Dodgers' farm system has been since the mid-2000s, when it was consistently in the Top 5 of system rankings. Adding to the Big Three with high-ceiling prospects helped the overall grade, and the depth has improved noticeably. I expect the system to remain strong. But if the front office decides to trade from it, they'll do it intelligently — either trading top prospects for true impact players or trading fringy guys for fringy upgrades. This system should be regarded as a Top 10 system in the game by most publications.
And a breakdown of the entire Top 100, by position
Player Position
LF/RF 16
RHSP 22
LHSP 3
RHRP 19
LHRP 7
Coming later today: Prospects 100-76
About Dustin Nosler
Dustin Nosler began writing about the Dodgers in July 2009 at his blog, Feelin' Kinda Blue. He co-hosts a weekly podcast with Jared Massey called Dugout Blues. He is a contributor/editor at The Hardball Times. He graduated from California State University, Sacramento, with his bachelor's degree in journalism and a minor in digital media. While at CSUS, he worked for the student-run newspaper The State Hornet for three years, culminating with a 1-year term as editor-in-chief. He resides in Stockton, Calif.
@DustinNosler
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Copyright © 2017 | Dodgers Digest | All Rights Reserved


text:
Eagle Eye: Sean Lee's Impact On Dallas 'D'
For the final installment of this week's Eagle Eye in the Sky, let's take a look at the Cowboys' defense and their performance against New York. If you missed it, check out Thursday's piece featuring the breakdown of the Dallas offense without Dez Bryant in the lineup.
After being one of the worst defenses over the past couple of seasons, I came away pretty impressed with Dallas against New York. Sure, the Giants' offense wasn't exactly firing on all cylinders, but the longest pass play Dallas allowed came on the final play of the game in a prevent defense. They've added pieces here and there, and despite the fact that some of their best players won't suit up on Sunday (Orlando Scandrick is on Injured Reserve; Randy Gregory is injured; and Rolando McClain and Greg Hardy are both suspended), they're a well-coached unit with talent at all three levels.
Along the defensive line, I thought Gregory flashed in his first game last week against New York. When he and Hardy return, they'll have a pretty good rotation with DeMarcus Lawrence, Jeremy Mincey and rookie Ryan Russell. A former starter at defensive end, Tyrone Crawford, slides inside to defensive tackle this season, and he had a big impact on the game against the Giants.
Crawford is athletic. He's disruptive and for a team that needed someone to step up inside he'll be a good fit as that penetrating 3-technique in Rod Marinelli's defense.
At the next level, you've got SAM linebacker Kyle Wilber (who will also rotate down to defensive end on some passing downs), second-year linebacker Anthony Hitchens (a player whom I liked a lot coming out of Iowa) and veteran Sean Lee has returned from injury to play at a high level. Lee missed all of 2014, but the former Nittany Lion looks like he's picked up right where he left off.
Lee looked instinctive as ever. He made athletic plays downhill and in coverage, and he was physical against the run as well. I showed a play where you saw him drop back in underneath zone coverage against the tight end. Dallas covered the New York tight ends a number of ways on Sunday night, and using Lee was just the start.
Here is the key play of the game. Third-and-goal, with the ability to put the game away, many thought the Giants would run the ball. But instead they chose to pass, with tight end Daniel Fells running a corner route in the end zone. Safety Barry Church refused to let Fells off the line of scrimmage though, and Manning was forced to throw the ball away, stopping the clock and giving the Cowboys the ability to come back late in the game. Church was charged with playing over the tight end when Dallas was in base packages, but there was one other trend I noticed as the game went on as well.
In the games I charted of the Cowboys' defense a year ago, they played just three percent of their snaps in a dime package, with six defensive backs on the field. They used the nickel package (five DBs) a lot, but very rarely did I see the dime package.
Last week, Dallas played in dime on 19 percent of its snaps, a huge difference from 2014. All of their snaps in dime came on third down, and they did a good job mixing up coverage concepts and they weren't predictable at all. The one thing that stayed consistent, however, was that rookie corner Byron Jones played over the tight end. Whether the Cowboys were in man or zone, it was Jones who lined up over Fells or Larry Donnell.
On that play, Jones did a great job forcing the incomplete pass against Donnell and bringing up fourth down. With players like Brent Celek and Zach Ertz giving the Cowboys trouble in the past, I would expect to see a mixture of Lee, Church and Jones playing against both of those Eagles tight ends on Sunday afternoon.
Fran Duffy is the producer of "Eagles Game Plan" which can be seen on Saturdays during the season. Be sure to also check out the "Eagle Eye In The Sky" podcast on the Philadelphia Eagles podcast channel on iTunes. Prior to joining the Eagles in 2011, Duffy was the head video coordinator for the Temple University Football team under former head coach Al Golden. In that role, he spent thousands of hours shooting, logging and assisting with the breakdown of the All-22 film from the team's games, practices and opponents.
Post-Snap Read: Jalen Hurts takes major step forward in his development
Fran Duffy highlights how the Eagles needed more from the quarterback in the pass game to get the win over Washington.
Post-Snap Read: Jonathan Gannon throws a curveball with the D-line
Fran Duffy uses the All-22 to highlight some of the tweaks that the Eagles made Sunday against the Giants.
Post-Snap Read: Not just a 'monster,' Dallas Goedert is one of the most complete tight ends in football
Fran Duffy looks at how Goedert not only shined on Sunday in the win over the Jets, but has been one of the best at his position all season.
Post-Snap Read: Watching Lane Johnson and Jordan Mailata dominate is a thing of beauty
The offensive line utilized its outstanding personnel and mixed in a variety of schemes to move the ball on the ground against the Giants.
Post-Snap Read: Stats alone don't do the rushing attack justice
Fran Duffy dives into the film to highlight the Eagles' dominant performance against what was previously the top-ranked run defense. It's a collaboration of players, coaching, and scheme that make it come to life.
Post-Snap Read: A film breakdown of Jalen Hurts' outstanding performance vs. Broncos
Plus, learn how the coaches used an offensive tendency to keep the Broncos off balance.
Post-Snap Read: How the rushing attack has fueled the offense
The Eagles' commitment to the run game has had a trickle-down effect on the rest of the offense. Fran Duffy takes a deep dive by pairing film and advanced statistics.
Post-Snap Read: Generating pressure leads to big day for defense
The Eagles disrupted the Lions' offense all game long thanks to a mix of blitzed and four-man rushes.
Post-Snap Read: Analyzing Dallas Goedert's debut as TE1
Goedert helped the offense in more ways than what showed up on the stat sheet Sunday.
Post-Snap Read: An ode to Zach Ertz, one of Philadelphia's all-time great athletes
Fran Duffy uses the film to highlight what made the three-time Pro Bowl tight end so special.
Post-Snap Read: Big-time plays from all levels of the defense
From Fletcher Cox and Javon Hargrave up front to Darius Slay and Steven Nelson on the back end to T.J. Edwards at linebacker, there were plenty of reasons for Sunday's outstanding performance.
Post-Snap Read: Tight ends remain a vital part of the offense
Yet, the Eagles are in 12 personnel less than previous seasons. Fran Duffy explains how the coaches are working in two of the offense's best players in Zach Ertz and Dallas Goedert.


Social media isn't all about sharing photos of your pets and 'liking' news of your long-lost school friend's.
If you know what you're doing social media can bring in some serious results to your business no matter what size you are.
Now, not every social media platform is right for every business. But the right social media platform can change your business for the better.
Whether it's the giant of Facebook or the lesser known Quora, you need to be on the platform that's right for you and make it apart of your strategy and purpose.
But why are businesses having trouble realizing the value of modern-day social media?
1. Businesses Find The Effectiveness Once They Start.
The businesses that invest in social media ads are more than twice as likely to say social media marketing is "very effective" for their business.
So the 18% who are unsure really have just not tried social media.
2. Social Media is More Affordable.
Social Media Advertising is a drop in the bucket compared to traditional outlets.
The monthly ad spend for the number of people you can reach and the increased traffic generated to a website is very cost effective and manageable for most businesses budgets.
In fact, to reach 1,000 people on Facebook you only have to give up $.25 cents.
3. Social Media Increases Website Traffic and Sales.
Social media sends customers to your store and to your online website.
It has actually been so powerful that marketers, brands, and businesses have started to allocate the biggest budget for social.
In fact, 92% of small businesses plan to invest more time and money in social media in 2018.
Most small businesses are going to increase their investment on Facebook, and the recent Cambridge Analytica scandal isn't slowing them down.
4. Social Media Creates a Better Customer Experience.
It is imperative that you are active on the Social Sites and available to answer questions, respond to comments and thank your customers for their reviews.
Well, social media is the first place users go to get customer support.
And if you don't have social media then you can expect some customers to have a bad experience.
In fact, 66% of consumers, worldwide, stopped doing business with companies due to poorly handled customer service.
But if you do decide to use social media as a customer service outlet, you can expect a great increase in sales.
The numbers speak for themselves. Customers want your business on Social Media, and consumers want to see that your business is consistent and active with Social Media.
5. What happens if you don't use social media?
Well, you will be left behind and the ones who do start using social media will survive.
To really put everything in perspective.
Nearly 3/4 of online adults use social media.
93% of consumers are researching social media before they make purchasing decisions.
41% of consumers who just spend 2 hours or less on social media still made on average $500 in purchases in the past 12 months through social.
74% of consumers rely on Social Media for more information or reviews about the products/services they want to purchase.
50% of consumers made a purchase based on a recommendation made on Social Media.
Social media for business is no longer optional, it is a must.
Focus on what works and what needs improvement to effectively bring in those desired results.
Where are you focusing your current marketing efforts?


It all started 13 years ago when someone had the zany idea to create an event featuring hand-crafted outhouses (complete with toilet seats!) on top of a pair of skis — and then race them in a daring downhill competition!
... both wait for crashes and hope they don't happen! Afterall, sitting in a toppling outhouse that is moving seems like it would be a nightmare.
Spectators come from all over the Southeast to get a look at the creative outhouses in the race. They can be built of anything, even cardboard!
Would you ever ride an outhouse on skis in a downhill race?
It seems a little daring, perhaps even dangerous!
Get a peek at the races and a look at what it's like to be inside one of these crazy downhill toilets by viewing the short video below from youtube user VisitSapphireValley.
The event takes place at Sapphire Valley Ski Area on Saturday, February 16, 2019 from 11:00 a.m. to 3:00 p.m. Admission to view the event is free!
Sapphire Valley Ski Area is located at 127 Cherokee Tr., Sapphire Valley, NC 28774. Learn more on their official website!


text:
package org.alfresco.contentcraft.command.macro;

import java.util.ArrayList;
import java.util.HashMap;
import java.util.Iterator;
import java.util.List;
import java.util.Map;

import org.bukkit.Location;
import org.bukkit.Material;
import org.bukkit.block.Block;
import org.bukkit.block.BlockFace;
import org.bukkit.event.block.BlockBreakEvent;
import org.bukkit.event.block.BlockPlaceEvent;
import org.json.simple.JSONArray;
import org.json.simple.JSONObject;

/**
 *
 * @author Roy Wetherall
 * @since 1.0
 */
public class Macro
{
	private String name;
	private boolean transientMacro = false;
	private Location startLocation;
	private boolean recording = false;
	private boolean runPending = false;
	private boolean runRepeat = false;
	private List<MacroAction> actions = new ArrayList<MacroAction>(27);

	/**
	 * Helper method that starts a new macro.
	 *
	 * @param name				name of the macro
	 * @return {@link Macro} 	newly created and started macro
	 */
	public static final Macro startNew(String name)
	{
		Macro macro = new Macro(name);
		macro.start();
		return macro;
	}

	/**
	 * Default constructor.
	 *
	 * @param name	name of the macro
	 */
	public Macro(String name)
	{
		this.name = name;
	}

	/**
	 * @return	{@link String}	name of the macro
	 */
	public String getName()
	{
		return name;
	}

	/**
	 * @return true if transient (ie not applicable for user to apply via command), false otherwise
	 */
	public boolean isTransient()
	{
		return transientMacro;
	}

	public List<MacroAction> getActions()
	{
		return actions;
	}

	public void runPending(boolean runRepeat)
	{
		if (!runPending)
		{
			if (recording)
			{
				// end recording
				recording = false;
			}

			runPending = true;
			this.runRepeat = runRepeat;
		}
	}

	public void run(Location location)
	{
		run(location, null);
	}

	public void run(Location location, MacroCallback callback)
	{
		for (MacroAction action : actions)
		{
			action.execute(location, callback);
		}
	}

	public void clear()
	{
		actions = new ArrayList<MacroAction>(27);
		startLocation = null;
	}

	public void start()
	{
		if (!recording)
		{
			recording = true;
			clear();
		}
	}

	public void stop()
	{
		recording = false;
		runPending = false;
		runRepeat = false;
	}

	/*package*/ void onBlockBreak(BlockBreakEvent event)
	{
		Block brokenBlock = event.getBlock();
		Location location = brokenBlock.getLocation();

		if (recording)
		{
			if (startLocation == null)
			{
				// record start location
				startLocation = location;
			}

			actions.add(BreakBlockMacroAction.create(brokenBlock, startLocation));
		}
	}

	/*package*/ void onBlockPlace(BlockPlaceEvent event)
	{
		Block placedBlock = event.getBlockPlaced();
		Location placedLocation = placedBlock.getLocation();

		if (recording)
		{
			if (startLocation == null)
			{
				// record start location
				startLocation = placedLocation;
			}

			actions.add(PlaceBlockMacroAction.create(placedBlock, startLocation));

			if (Material.WOODEN_DOOR.equals(placedBlock.getType()) ||
				Material.IRON_DOOR.equals(placedBlock.getType()))
			{
				System.out.println("Dealing with top of the door");
				// need to record the top part of the door too
				Block topDoor = placedBlock.getRelative(BlockFace.UP);
				actions.add(PlaceBlockMacroAction.create(topDoor, startLocation));
			}
		}
		else if (runPending)
		{
			if (!runRepeat)
			{
				runPending = false;
			}
			run(placedLocation);
		}
	}

	@SuppressWarnings("unchecked")
	/*package*/JSONObject toJSON()
	{
		JSONObject jsonMacro = new JSONObject();
		jsonMacro.put("name", getName());

		JSONArray jsonActions = new JSONArray();
		for (MacroAction action : getActions())
		{
			jsonActions.add(action.toJSON());
		}

		jsonMacro.put("actions", jsonActions);
		return jsonMacro;
	}

	@SuppressWarnings("rawtypes")
	/*package*/ static Macro fromJSON(JSONObject jsonMacro)
	{
		Macro macro = new Macro((String)jsonMacro.get("name"));

		if (jsonMacro.containsKey("transient"))
		{
			boolean isTransient = (Boolean)jsonMacro.get("transient");
			macro.transientMacro = isTransient;
		}

		JSONArray jsonActions = (JSONArray)jsonMacro.get("actions");
		try
		{
			Iterator it = jsonActions.iterator();
			while (it.hasNext())
			{
				JSONObject jsonAction = (JSONObject)it.next();

				String actionType = (String)jsonAction.get("type");
				Class actionClass = Class.forName(actionType);
				MacroAction action = (MacroAction)actionClass.newInstance();

				action.fromJSON(jsonAction);
				macro.actions.add(action);
			}
		}
		catch (Exception e)
		{
			e.printStackTrace();
		}

		return macro;
	}
}

redpajama_set_name: RedPajamaGithub


Focusing on one side of your body at a time can make your whole program even more effective.
If your idea of unilateral training is to toss a few single-arm or single-leg exercises into your workout as an afterthought, you're probably not reaping their full strength-building, injury-proofing benefits.
Most (if not all) of us are asymmetrical, meaning one limb will almost always be bigger and stronger than the other. That muscle imbalances become most obvious when one limb has to support weight on its own during those unilateral movements (think: single-leg squats and one-arm presses), but if they're too extreme, the disparities can sneak into bilateral exercises as well (think: back squats and pushups).
For example, even if both arms have to work to push you up off the floor, you're probably using more of one than the other—without even realizing it. "It's very rare for somebody to have 50-50 weight distribution because almost everyone has a dominant limb," says Joel Seedman, Ph.D., owner of Advanced Human Performance and author of Movement Redefined.
You'll probably never achieve perfect symmetry between both sides of your body, unfortunately—that's just part of our makeup. That said, you should try to get as close to even as possible, especially if you're shooting for a developed physique. "The more symmetrical we are, the fewer imbalances and compensation patterns we have," Seedman explains. The result: Greater overall strength and efficiency, as well as a reduced likelihood of injury.
Take the back squat for example. When you squat with muscle imbalances, your chance of injuring your back, knees or hips is markedly increased. Instead of generating force straight up through your core, that force pulls on the body in uneven positions and directions, according to Seedman.
Similarly, running with strength imbalances can up your risk of injury. A 2005 study published in Clinical Journal of Sport Medicine found that runners with single-leg overuse injuries had noticeable strength imbalances between the hip muscles on their injured and non-injured side. Whether the injuries were the cause or the result of the muscle imbalances is unknown—but researchers suggest these findings show a relationship between the two.
If you can eliminate those imbalances, each limb will be able to handle weight more evenly, which only helps make you stronger and more efficient in bilateral movements, Seedman says.
You can try to even yourself out during bilateral exercises, but it's pretty hard to get both sides to fire equally unless each is forced to learn how to fend for itself. "Otherwise, you're going to revert back to those compensation patterns," Seedman says. So, if you care about getting stronger and minimizing your injury risk, unilateral exercises (like single-leg squats, single-leg Romanian deadlifts and one-arm presses) are a must.
Bonus: Most guys don't realize that single-arm exercises in particular do double-duty as major core-builders. If you do a single-arm chest press, for example, your core has to fire like crazy to keep you from rolling off the bench, according to Seedman. One of his favorite core-crushing, symmetry-building moves is the single-arm eccentric accentuated dumbbell push press.
Keep this advice in mind next time you hit the gym, and your gains—and muscle balance—will soon follow. Instead of blowing through a set of lunges or single-arm presses and calling it a day, incorporate these tips to maximize strength and stability.
"Some people just kind of assume you do the heavy bilateral lift first and then you do the unilateral, but there's no reason you can't do it the other way around," says celebrity trainer Ben Bruno.
Starting your workout with unilateral exercises is actually a great way to activate muscles that would normally be a little lazy during your bilateral exercises. Sure, you run the risk of fatiguing your muscles before your big lift, but Seedman thinks there could be some warmup element at play. " You may actually be able to lift more weight because the muscles are firing better and more symmetrically," he says.
Even better, Seedman believes you'll be less likely to fall back into old compensation patterns where your stronger side picks up the slack.
Lunges and Bulgarian split squats are great, but you can make the argument that they're not 'true' unilateral exercises.
So, be sure to include 'true' unilateral exercises in your routine, like single-leg Romanian deadlifts, single-leg squats, skater squats, step-ups and single-leg hip thrusts, to name a few.
Bruno recommends treating each group of reps using each limb as an individual set—especially when you go heavy. "Anybody who's done a heavy set of lunges or Bulgarian split squats knows it's very taxing," he says.
If you move straight from one side to the other when you're still huffing and puffing, your form might break down before you end your set, and you won't reap the full strength and stability benefits. Don't hesitate to hit pause if you need to. "I think it just helps to take a little break, both from a breathing standpoint, and if you're holding dumbbells, from a grip standpoint," Bruno says.
According to Seedman, you'll probably only need 15 seconds of rest between sides—but if you're going heavy or you're especially fatigued, you may need a full minute.
Unilateral exercises—especially single-leg—can be humbling, because you probably won't be able to use a ton of weight at first. But if you go too heavy too soon, Bruno warns that you'll have a hard time keeping your balance.
Resist the urge to load up your single-leg squats and deadlifts right away. Your first priority is to master the move and get used to balancing on one leg. Then, you can start adding weight.