Example 5. Calculate the integral

$$
\int_{0}^{i} z \cos z d z
$$
Solution. The functions $f(z)=z$ and $\varphi(z)=\cos z$ are analytic everywhere. Applying the formula for integration by parts, we get

$$
\begin{aligned}
\int_{0}^{i} z \cos z d z & =\int_{0}^{i} z(\sin z)^{\prime} d z=\left.(z \sin z)\right|_{0} ^{i}-\int_{0}^{i} \sin z d z= \\
& =i \sin i+\left.\cos z\right|_{0} ^{i}=-\operatorname{sh} 1+\operatorname{ch} 1-1=\frac{1-e}{e}
\end{aligned}
$$

## Single-valued branches of a multi-valued function.

Branch points

Suppose the function $w=f(z)$, analytic in the domain $D$, maps $D$ onto the domain $G$ and such that the inverse function $z=\varphi(w)$ is multi-valued in the domain $G$. If there exist single-valued, analytic in the domain $G$ functions $z=\varphi_{1}(w), z=\varphi_{2}(w), \ldots$, for which the given function $w=f(z)$ is the inverse, then the functions $\varphi_{1}(w), \varphi_{2}(w), \ldots$ are called single-valued branches of the function $\varphi(w)$, defined in the domain $G$.

For example, the function $w=z^{n}$ assigns a unique point $w_{0}$ to each point $z_{0}$, but the same point $w_{0}(w \neq 0, w \neq \infty)$ is assigned $n$ different points in the $z$-plane by the function $z=\sqrt[n]{w}$; in this case, if $w=\rho e^{i \theta}$, then these $n$ values of $z$ are given by the formulas $z_{k}=r e^{i \varphi_{k}}$, where

$$
r=\sqrt[n]{\rho}, \quad \varphi_{k}=\frac{\theta}{n}+\frac{2 k \pi}{n} \quad(-\pi<\theta \leqslant \pi, k=0,1,2, \ldots, n-1)
$$

Let a simply connected domain $G$ contain the point $w_{0}$ but not contain the points $w=0$ and $w=\infty$. Then different fixed values of $k(k=0,1,2, \ldots, n-1)$ for the same choice of the number $\theta_{0}$ (for example, $\theta_{0}=\arg w_{0}$) correspond to different branches of the function $z=\sqrt[n]{w}$.

A point that has the property that a circuit around it in a sufficiently small neighborhood leads to a transition from one branch of a multi-valued function to another is called a branch point of the considered multi-valued function. The branch points of the function $\sqrt[n]{w}$ are the points $w=0$ and $w=\infty$.

After $n$ circuits around the point $w=0$, we return to the original branch of the function $\sqrt[n]{w}$; branch points with such a property are called algebraic branch points of order $\boldsymbol{n}-1$. At each of these points, the function has only one value: $\sqrt[n]{0}=0, \sqrt[n]{\infty}=\infty$, i.e., the branches of the function at these points coincide.

For the logarithmic function $w=\operatorname{Ln} z$, the branch points are $z=0$ and $z=\infty$, and $\ln 0=\infty$ and $\ln \infty=\infty$. Any finite number of circuits (in the same direction) around the point $z=0$ will not return to the original branch of the function $\operatorname{Ln} z$. Such branch points are called logarithmic. When integrating, it is necessary to specify a branch of the multi-valued function. This is achieved by specifying the value of the multi-valued function at some point on the integration contour.

If the integration contour $C$ is closed, then the initial point $z_{0}$ of the integration path is considered to be the point where the value of the integrand is specified.