text:
Airport Jobs in St Lucia
Average Salary: $105,000
Salary from from $0 from $5,000 from $10,000 from $15,000 from $20,000 from $25,000 from $30,000 from $35,000 from $40,000 from $45,000 from $50,000 from $55,000 from $60,000 from $65,000 from $70,000 from $75,000 from $80,000 from $85,000 from $90,000 from $95,000 from $100,000 from $105,000 from $110,000 from $115,000 from $120,000 from $125,000 from $130,000 from $135,000 from $140,000 from $145,000 from $150,000
Salary to to $5,000 to $10,000 to $15,000 to $20,000 to $25,000 to $30,000 to $35,000 to $40,000 to $45,000 to $50,000 to $55,000 to $60,000 to $65,000 to $70,000 to $75,000 to $80,000 to $85,000 to $90,000 to $95,000 to $100,000 to $105,000 to $110,000 to $115,000 to $120,000 to $125,000 to $130,000 to $135,000 to $140,000 to $145,000 to $150,000+
Brisbane Region
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Accounting & Finance Jobs (1)
Health Workforce Queensland (8)
governmentdefencework from homecouncilno experienceagricultureagriculturalpoliceinvestigatorfunmore ❯
Find a job this new financial year 30 JUN
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Occupational Therapist - Mackay/Whitsundays
NEGOTIABLE -
HEALTH WORKFORCE QUEENSLAND
, and Melbourne (Virgin Australia, Jetstar, Qantas, Tiger) from Proserpine Airport, Hamilton Island Airport, Mackay Airport Position Description: ATTENTION Wish you worked here? This fabulous ... More details ❯
Financial Accountant
$100/110,000 + SUPERANNUATION -
HAYS-
CAIRNS, QUEENSLAND
A leadership position within an established, inclusive culture. Head Office function in Cairns. Your new company Cairns Airport is one of Australia's leading regional airports ... , providing air links to a variety of regional, domestic and international destinations, Pre Covid Cairns Airport welcomed over 5 million passengers a year. The Airport is owned and operated ... More details ❯
General Practitioner Mackay
... plenty of amenities, including excellent schools, shopping centres, cinemas, sporting facilities & activities, beaches, cafes and restaurants. The Mackay airport has ample flight ... More details ❯
It takes 2 minutes and it 's free.
General Practitioner - Mackay
... and its own University it also has many extracurricular activities available. It has its own airport with the three major airlines and is less than a two-hour drive to the Whitsundays ... More details ❯
NORTH MACKAY
... University it also has many extracurricular activities available. It has its own airport with the three major airlines and is less than a two-hour drive to the Whitsundays and Hamilton ... More details ❯
Podiatrist - Toowoomba
. With our own Wellcamp (Brisbane West) Airport we have easy domestic connections to Sydney, Melbourne, Northern Queensland, Northern Territory and international. Toowoomba has ... More details ❯
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Registered Nurse - Yungaburra
YUNGABURRA
... an international airport with regular flights throughout Australia and the world, a campus of James Cook University, easy access to the Great Barrier Reef and great shopping. The following ... More details ❯
General Practitioner - Sarina
... a tropical paradise. Six picturesque sandy beaches are in close proximity to town. Mackay airport is 35 km away with 11 flights into Mackay from Brisbane and 11 from Mackay to Brisbane ... More details ❯
General Practitioner - Bundaberg
... has a culturally diverse population of 93,000 and is the 9th most populous city in Queensland. It has three hospitals, an airport, private and public schools, a university campus ... More details ❯
Ramp Officer
TOWNSVILLE, QUEENSLAND
... working out of the Townsville airport. Your new role Your new role will see you responsible for safe baggage and freight handling of goods such as general packages, live animals ... More details ❯
Jobs ❯ Queensland ❯ St Lucia ❯ Other/General Jobs in St Lucia ❯ Airport Jobs in St Lucia
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text:
Q: Creating plots with disconnected/discrete lines Here is my code:
for p = 1:length(id_unique)
    h=figure;
    hold on
    j = 1;
    for i = 1:1:length(id)
        if id_unique(p)==id(i)
            h=plot([startdate(i),enddate(i)],[j,j],'ok-');
            hold on
            j = j + 1;
        end
    end
    grid on
    hold off
    savefig([plotname,'.fig'])
    print(plotname,'-djpeg','-r300')
    close
end

% id:           integer vector containing 50000 values
% id_unique:    sorted unique values from vector 'id'
% startdate:    datetime vector containing 50000 dates
% enddate:      datetime vector containing 50000 dates

Each elements/values in 'id' vector implies an event where the startdate and the enddate of the event is available in the corresponding position in 'startdate' and 'enddate' vectors.
Thus an event id(1) has the start date in startdate(i) and end date in enddate(i).
The program takes a value from the 'id_unique' vector and for each matched value found in 'id', it draws a line in the plot, denoting the beginning and ending time of the event.
For example, suppose 55 is an id-value from vector 'id_unique' and we have this value 1000 times in id. So for 55 a plot is created depicting 1000 separate lines with a marker 'o' at the beginning of the event, a marker 'o' at the end of the event and a line connecting both markers.

Please look at the attached plot that is generated from this code block. If id_unique has 70 values, 70 such plots will be created from this code. In the image, many lines are too small because of small difference between start and end date, so the markers overlap each other and look like a single point.
Now the problem comes when for an id-value in 'id_unique', we have a lot of instances of it in the 'id' vector. When the program plots individual lines unto 100 it works quite fast, but after plotting 300 lines in the same figure the program gets slow. When the program plots 1000 lines in the same figure, it takes about 5-7 seconds for each line. So it takes many hours to generate a plot with many lines.
Is there a way to improve my code to make these plot generation faster.

A: You don't need a loop:
you can use something like:
a = 1:0.1:4*pi;
b = sin(a); %vector that represent the start of each line
c = sin(a)+1; %vector that represent the end of each line
plot([b;c],[a;a],'-ko','markersize',2)

Result:

The function plot need 2 arguments: x and y, but, and this is why the world is beautiful, the function plot can manage several lines. If x and y are matrices, matlab interpret each column as a new line.

A: Having thought about this more, I now would suggest doing the following - rather than creating multiple lines in a plot, which generates a lot of objects in the background, it is better to produce one line per plot, that is broken up into multiple segments.  If startdate and enddate are vectors of 1xlength(id), then the following code will be much faster as it concatenates all of the lines together, with NaN values in between.  This makes the plot break the line at that point.  All very quick, without the need to mess about with xlim and ylim as I previously suggested...
Here's my code:
for p = 1:length(id_unique)
    h=figure;
    block=[startdate(id==id_unique(p));enddate(id==id_unique(p))];
    x_vals=reshape([block;nan(1,size(block,2))],1,[]);
    y_vals=reshape([(1:size(block,2));(1:size(block,2));nan(1,size(block,2))],1,[]);
    plot(x_vals,y_vals,'ok-');
    grid on
    savefig([plotname,'.fig'])
    print(plotname,'-djpeg','-r300')
    close
end

I hope this works better for you.  (Of course, if your startdate and enddate vectors are length(id) by 1, then you should transpose them using .' in the 3rd line above).
I'm not sure what the purpose of the separate plots is, but it would be possible to plot them all on the same graph if that were useful.  The code would then be:
h=figure;
hold on;
for p = 1:length(id_unique)
    block=[startdate(id==id_unique(p));enddate(id==id_unique(p))];
    x_vals=reshape([block;nan(1,size(block,2))],1,[]);
    y_vals=reshape([(1:size(block,2));(1:size(block,2));nan(1,size(block,2))],1,[]);
    plot(x_vals,y_vals,'Marker','o');
    grid on
end
savefig([plotname,'.fig'])
print(plotname,'-djpeg','-r300')
close

This lets Matlab set the line type and colour using its standard series.  It also allows you to use the legend function to add labels.  Of course, if the position within id were important, rather than using a sequence for y_vals, you could use the position information obtained from find, changing the 4th and 6th lines to be:
[~,index]=find(id==id_unique(p));
block=[startdate(index);enddate(index)];

y_vals=reshape(index;index;nan(1,size(block,2))],1,[]);

Then you see all of id on one plot, with the different values of id distinguished by colour and linetype.  You can then use the legend function to generate a legend:
legend(num2str(id_unique.'));

(This assumes that id_unique is a row vector.  If it is a column vector, remove the .').

A: Here is way to also plot 2 different markers:
% some random data:
N = 50;
id = randi(5,N,1);
startdate = sort(randi(100,N,1));
enddate = startdate+randi(10,N,1);

% plotting:
ax = plot([startdate(:).'; enddate(:).'],[1:N; 1:N],'-k',...
    startdate,1:N,'k<',enddate,1:N,'k>')

The (:).' after the vectors is not really  needed, it's just to make sure they will be given in the right direction to plot - 2 rows, startdate above enddate.
which gives (randomly):

If you want to divide the data to groups by id, and color them this way, you can do the following:
N = 30;
id = randi(5,N,1);
startdate = datetime(sort(736000+randi(100,N,1)),'ConvertFrom','datenum');
enddate = startdate+randi(20,N,1);

% plotting:
ax = plot([startdate(:).'; enddate(:).'],[1:N; 1:N],'-',...
    startdate,1:N,'k<',enddate,1:N,'k>');

% coloring:
cmap = colormap('jet');
col = cmap(floor(linspace(1,64,numel(unique(id)))),:);
for k = 1:N
    ax(k).Color = col(id(k),:);
    ax(k).LineWidth = 2;
end

% set the legend:
c = 1;
leg_ent = zeros(numel(unique(id)),1);
for k = unique(id).'
    leg_ent(c) = find(id==k,1,'first');
    c = c+1;
end
legend(ax(leg_ent),num2str(unique(id)),'Location','SouthEast')

and you will get:

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text:
\section{\bf 2. Geometry and eigenforms.}

The geometry has been outlined in Cheeger [\putref{cheeg1}] and already
used in our
earlier work, [\putref{BKD}], so that it will not be described at length
here. The cone manifold, ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$ has the hyperspherical polar metric
$ds^2=dr^2+r^2d\Sigma^2$ with $d\Sigma^2$ the metric on the base, ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$,
which is
the $r=1$ section of the cone. For simplicity, we shall assume in this
paper
that the base is closed, $\partial{\cal N}}\def\nwilde{{\widetilde{\cal N}}=\emptyset$, which is enough for the ball.

The structure of the eigenforms of the de Rham Laplacian, $\Delta_{\cal M}}\def\mwilde{{\widetilde{\cal M}}$,
on the infinite generalised cone has been given by Cheeger [\putref{cheeg1}]
and, for convenience, we refer to [\putref{cheeg1}] for the detailed action
of $\Delta_{\cal M}}\def\mwilde{{\widetilde{\cal M}}$ on forms of the type $\beta+dr\wedge\omega$ as well as for the
necessary exterior calculus. (See also [\putref{cheeg2}].)

There are four basic types of eigenforms with nonzero eigenvalues, $\alpha^2$,
$$
\phi^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}(1)}_p={J_{\nu(p)}(\alpha r)\over r^{(d-1-2p)/2}}\,\phi^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_p\,,
\eql{eigform1}
$$

$$
\phi^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}(2)}_p={J_{\nu(p-1)}(\alpha r)\over r^{(d+1-2p)/2}}\,
\tilde d\phi^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_{p-1}+\bigg({J_{\nu(p-1)}(\alpha r)\over r^{(d+1-2p)/2}}
\bigg)'\,dr\wedge \phi^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_{p-1},
\eql{eigform2}$$

$$\eqalign{
\phi^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}(3)}_p={1\over r^{d-2p}}\big(r^{(d+1-2p)/2}\,&J_{\nu(p-1)}
(\alpha r)\big)'\tilde d\phi^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_{p-1}\cr+
&{J_{\nu(p-1)}(\alpha r)\over r^{(d+3-2p)/2}}dr\wedge
\tilde\delta\tilde d\phi^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_{p-1},\cr}
\eql{eigform3}$$

$$
\phi^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}(4)}_p={J_{\nu(p-2)}(\alpha r)\over r^{(d+1-2p)/2}}\,dr\wedge
\tilde d\phi^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_{p-2},
\eql{eigform4}$$
where $\phi^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_p$ is a coexact $p$-eigenform of the intrinsic
de Rham Laplacian, $\Delta_{\cal N}}\def\nwilde{{\widetilde{\cal N}}=\tilde d\tilde\delta+\tilde\delta\tilde d$,
on the base ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$. We have the separation of variables relation
$$
\nu(p)=\big(\mu(p)+((d-1)/2-p)^2\big)^{1/2}
\eql{reln}$$ with $\mu(p)$ being the (coexact) eigenvalue of $\Delta_{\cal N}}\def\nwilde{{\widetilde{\cal N}}$.
We are assuming that $\nu\ge1$ so that the `negative' modes
$\sim J_{-\nu}$ do not arise.

In addition, there are modes (`zero modes') whose ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$ part is harmonic,
$\mu=0$,
$$
\phi^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}(E)}_p={J_{\nu_E(p)}(\alpha r)\over r^{(d-1-2p)/2}}\,h^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_p
\eql{eigformE}
$$
and
$$
\phi^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}(O)}_p=\bigg({J_{\nu_O(p)}(\alpha r)\over r^{(d+1-2p)/2}}
\bigg)'\,dr\wedge h^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_{p-1},
\eql{eigformO}$$
with
$$\eqalign{
\nu_E(p)&=|(d-1)/2-p|=\nu_E(d-1-p)\cr
\nu_O(p)&=|(d+1)/2-p|=\nu_E(d-p).\cr}
\eql{znus}$$

On ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$, types 1, $E$ and 3 are coexact and types 2, $O$ and 4 are
exact. (See the relations in [\putref{cheeg2}].)

For the bounded, generalised cone, conditions are to be set at $r=1$.
Absolute boundary conditions are [\putref{cheeg3,Gilkey1}]
$$
\big(\phi^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}}_{i\ldots j}\big)'\bigg|_{\cal N}}\def\nwilde{{\widetilde{\cal N}}=0,\quad
\phi^{\cal M}}\def\mwilde{{\widetilde{\cal M}}_{r,i\ldots j}\bigg|_{\cal N}}\def\nwilde{{\widetilde{\cal N}}=0
\eql{absbc}$$
and have to be applied to the six types separately.

Since $\phi^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_p$, $h^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_p$ are pure ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$ forms, it is easily shown
that
types 1, 2, $E$ and $O$ satisfy Neumann (Robin) and types 3 and 4 Dirichlet
conditions. Bessel's equation has to be used to derive this for type 3.

More precisely the Robin conditions are ({\it e.g. } for type 1)
$$
\partial_r\big(r^{p-(d-1)/2}J_{\nu(p)}(\alpha r)\big)\bigg|_{\cal N}}\def\nwilde{{\widetilde{\cal N}}=0
\eql{arc}$$
so that the Robin parameter is $u=u_a(p)=p-(d-1)/2$,
([\putref{BGKE}] 3.1). The parameter for type 2 is $u_a(p-1)$.

Relative boundary conditions are obtained by dualising and, in the present
context amount to
$$
\big(\partial_r+d+2-2p\big)\phi^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}}_{r,i\ldots j}\bigg|_{\cal N}}\def\nwilde{{\widetilde{\cal N}}=0,
\quad\phi^{\cal M}}\def\mwilde{{\widetilde{\cal M}}_{i\ldots j}\bigg|_{\cal N}}\def\nwilde{{\widetilde{\cal N}}=0.
\eql{relbc}$$
Now types 1, 2, $E$ and $O$ satisfy Dirichlet and types 3 and 4 Robin
conditions. For type 3, Robin conditions read
$$
\partial_r\big(r^{(d+1)/2-p}J_{\nu(p-1)}(\alpha r)\big)\bigg|_{\cal N}}\def\nwilde{{\widetilde{\cal N}}=0
\eql{rrc}$$
with the Robin parameter $u=u_r(p)=(d+1)/2-p=u_a(d-p)$. The parameter for
type 4 is $u_r(p-1)$. Hodge $\star$-duality on ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$ thus interchanges
the conditions (\puteqn{arc}) and (\puteqn{rrc}) with $1\leftrightarrow4$
and $2\leftrightarrow3$ which differs from Cheeger.

We denote the coexact degeneracies on ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$ by $d(p)$ and remark that the
exact degeneracies, $d_{ex}(p)$, and eigenvalues, $\mu_{ex}(p)$, are given
by
$$
d_{ex}(p)=d(p-1),\quad
\mu_{ex}(p)=\mu(p-1).
\eql{degrel}$$
The structure of the eigenforms shows that the degeneracies of types
1, 2, 3 and 4 are $d(p)$, $d(p-1)$, $d(p-1)$ and $d(p-2)$ respectively.

We now define, for later use, the modified coexact $\zeta$--function\ on ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$,
the base $\zeta$--function,
$$
\zeta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_p(s)=\sum {d(p)\over\nu(p)^{2s}}=\sum {d(p)
\over\big(\mu(p)+((d-1)/2-p)^2\big)^s}
\eql{modcoexzet}$$
({\it cf } [\putref{BKD}] (3.3).) Including the `zero modes', which is sometimes
notationally convenient, one has
$$
\tilde\zeta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_p(s)=\zeta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_p(s)+{\beta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_p\over\nu_E(p)^{2s}}
\equiv\sum {\tilde d(p)\over\nu(p)^{2s}},
\eql{zetzeroin}$$
which defines $\tilde d(p)$ and where $\beta_p^{\cal N}}\def\nwilde{{\widetilde{\cal N}}$ is the $p$-th Betti
number of ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$.
The summations, here and later, are over the mode
labels which are not always explicitly displayed.

In view of the relations (\puteqn{degrel}), an `exact'
$\zeta$--function\ is defined by
$$
\zeta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_{ex,\,p}(s)=\zeta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_{p-1}(s).
\eql{modexzet}$$
Because the additional term $\big((d-1)/2-p\big)^2$ depends on $p$,
$\zeta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_{ex,\,p}(s)$ is not the true exact $\zeta$--function, but we expect it to be
the relevant construct.

Assume for the time being that the base ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$ has no boundary. Hodge
duality on ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$ can be applied to yield the coexact relations
$$
d(d-1-p)=d(p),\quad{\rm and}\quad \mu(d-1-p)=\mu(p)
\eql{dual1}$$
whence
$$
\nu(d-1-p)=\nu(p)
\eql{dual2}$$
and therefore, for the coexact $\zeta$--function,
$$
\zeta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_{d-1-p}(s)=\zeta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_p(s).
\eql{zedual}$$

These relations will be verified later when ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$ is a sphere.

\section{\bf 3. General constructions.}
The {\it total} $\zeta$--function\ on ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$ is, according to
(\puteqn{eigform1}) -- (\puteqn{eigform4}), (\puteqn{eigformE}), (\puteqn{eigformO}),
a combination of exact and coexact (on ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$) contributions.
Noting that the zero modes on ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$ give degenerate eigenvalues on ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$,
one has
$$
\zeta^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}+}_p(s)=\sum_{i=1}^4\sum_{\alpha_i}{1\over\alpha_i^{2s}}
+\beta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_p\sum_{\alpha_E}{1\over\alpha_E^{2s}}
+\beta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_{p-1}\sum_{\alpha_O}{1\over\alpha_O^{2s}}
\eql{totzetm}$$
where the $\alpha_i^{2s}$ are the eigenvalues of the $p$-form Laplacian on ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$
for $i$-type modes.

(\puteqn{totzetm}) can be written in terms of the coexact $\zeta$--function\ on ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$,
$\zeta^{\cal M}}\def\mwilde{{\widetilde{\cal M}}_p(s)$, as
$$
\zeta^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}+}_p(s)=\zeta^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}}_p(s)+\zeta^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}}_{p-1}(s),
\eql{totzet2}$$
the inverse of which is
$$
\zeta_p^{\cal M}}\def\mwilde{{\widetilde{\cal M}}(s)=\sum_{q=0}^p(-1)^{p-q}\zeta_p^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}+}(s).
\eql{inver}$$

According to the method given in [\putref{BEK,BGKE,BKD}],
and now for absolute conditions,
$$
\zeta^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}}_{a,p}(s)=\!\!\sum\!
\int_\gamma{dk\over2\pi i}k^{-2s}{\partial\over\partial k}\!
\bigg(\tilde d(p)\ln\big(r^{u_a(p)} J_{\nu(p)}(kr)\big)'\bigg|_{\cal N}}\def\nwilde{{\widetilde{\cal N}}
+d(p-1)\ln J_{\nu(p-1)}(k)\bigg),
\eql{coexze}$$
where the details of the contour $\gamma$ are in the above cited references.
The first term is the Neumann (Robin) (types 1 and $E$)
and the second term the Dirichlet (type 3) part.
For economy of writing, the degeneracy $\tilde d(p)$ is introduced so as
to take into account the $\beta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_p$ $E$-type zero modes on ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$,
{\it cf } (\puteqn{zetzeroin}). When $p\to p-1$, the type 1 contribution becomes
a type 2,  the type $E$ a type $O$ and the type 3 a type 4.

For relative conditions, likewise,
$$
\zeta^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}}_{r,p}(s)=\!\!\sum\!
\int_\gamma{dk\over2\pi i}k^{-2s}{\partial\over\partial k}
\bigg(d(p-1)\ln\big(r^{u_r(p)}J_{\nu(p-1)}(kr)\big)'\bigg|_{\cal N}}\def\nwilde{{\widetilde{\cal N}}
+\tilde d(p)\ln J_{\nu(p)}(k)\bigg).
\eql{rcoexze}$$

It is amusing to check in detail Hodge duality on ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$,
$$
\zeta^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}+}_{a,p}(s)=\zeta^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}+}_{r,d+1-p}(s),
\eql{hdual}$$
which fundamentally arises from the intertwining [\putref{Gilkey1}]
$$
\star\,\Delta^{\cal M}}\def\mwilde{{\widetilde{\cal M}}_{p,a}=\Delta^{\cal M}}\def\mwilde{{\widetilde{\cal M}}_{d+1-p,r}\star.
$$

It is easily seen from (\puteqn{totzet2}) that (\puteqn{hdual}) is equivalent
to the statement that, under $p\to d-p$, $\zeta^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}}_{a,p}(s)$ of
(\puteqn{coexze}) turns into $\zeta^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}}_{r,p}(s)$ of (\puteqn{rcoexze}).
Firstly, this is readily seen to be the case from the relations
(\puteqn{dual1}) and (\puteqn{dual2}), if the zero $E$ modes are set aside.
Secondly, consider the contribution coming from the
zero $E$ modes in (\puteqn{coexze}). One has $u_a (p) = p-(d-1)/2$ and
the argument of the logarithm reads explicitly
$
u_a (p) J_{\nu_E(p)} +kJ'_{\nu_E(p)}
$
where $|u_a (p)| = \nu_E(p)$. Let us assume for the moment that
$u_a(p)\leq 0$, and then use the relation
$
z J_{\nu}' (z) -\nu J_{\nu} (z) = - z J_{\nu+1} (z)
$
to write the above argument as
$
-k J_{\nu_E(p) +1}.
$

The factor $(-k)$ does not contribute because it has no zero
inside the contour, $\gamma$, and so the contribution
is just that of $J_{\nu_E(p) +1}$.
Now from (\puteqn{znus}), $ \nu_E (p) +1=\nu_E(d-p)$
and, taking into account Poincar\'e duality on ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$, $\beta_p^{\cal N}}\def\nwilde{{\widetilde{\cal N}}
=\beta_{d-p}^{\cal N}}\def\nwilde{{\widetilde{\cal N}}$, the contribution of the $p$-form
zero modes for absolute boundary conditions is seen to equal
that of the $(d-p)$-form zero modes for relative
boundary conditions, which was to be shown in order to complete the
demonstration of Hodge duality on ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$.
For $u_a (p) > 0$ use
$
z J_{\nu} ' (z) +\nu J_{\nu } (z) = z J_{\nu -1} (z)
$
to arrive at the same conclusion.

One of our main objectives is the evaluation of the
coefficients, $A_{n/2}$, in the heat-kernel expansion, which is stated in
generic form,
$$
K^{\cal M}}\def\mwilde{{\widetilde{\cal M}}(\tau)=\sum_{n=0}^\infty A_{n/2}^{\cal M}}\def\mwilde{{\widetilde{\cal M}}\,\tau^{(n-D)/2}+A'_{\cal M}}\def\mwilde{{\widetilde{\cal M}}
\log\tau.
\eql{expans}$$

Our preferred general computational formula,
on any manifold, ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$, of dimension $D$, is
$$
A^{\cal M}}\def\mwilde{{\widetilde{\cal M}}_{n/2}=\Gamma\big((D-n)/2\big){\rm Res\,}\zeta^{\cal M}}\def\mwilde{{\widetilde{\cal M}}\big((D-n)/2\big),
\quad n<D.
\eql{coefff}$$

The ability, by choosing $D$ sufficiently large, to work with just this
formula in order to determine {\it any} coefficient, has an additional
advantage because a finite number of `extra' modes does not affect the
analytic structure, as in the difference between $\zeta$ and $\tilde\zeta$ when
(\puteqn{coefff}) is applied to the base ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$.

The calculation of the $\zeta$--function\ on ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$ in its relation to that on ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$,
follows precisely the previous pattern, [\putref{BKD}], the Robin and
Dirichlet cases now being combined. It leads
to the following basic equation for the coexact heat-kernel coefficients
on ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$ in terms of those on ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$, ($n<d$),
$$\eqalign{
A^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}}_{a,n/2}(p)=&{1\over2\sqrt\pi(D-n)}\big({A}^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_{n/2}(p)
+A^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_{n/2}(p-1)\big)\cr
&+{1\over4}\big(A^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_{(n-1)/2}(p)-
A^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_{(n-1)/2}(p-1)\big)\cr
&-\sum_{i=1}^{n-1}\bigg(A^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_{(n-1-i)/2}(p)\,P_{i}\big(z_a(p)\big)
+A^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_{(n-1-i)/2}(p-1)\,P_i(x)\bigg)\cr}
\eql{comancoeff}$$
for absolute conditions.

The $A^{\cal N}}\def\nwilde{{\widetilde{\cal N}}$ are the heat-kernel coefficients corresponding to the
base $\zeta$--function, (\puteqn{zetzeroin}), and the $P_i$ are known polynomials arising
from the asymptotic expansion of Bessel functions, {\it e.g. }
$$
P_i(x)=\sum_{b=0}^i x_{i,b}\,
{\Gamma\left((D-n+i)/2 +b\right)\over
\Gamma \left((D-n+i)/2\right)\Gamma(b+i/2)}.
\eql{polys}$$

The $x$'s correspond to Dirichlet and the $z_a(p)$'s, which depend on $p$
through the $u_a(p)$, to Robin conditions.

The combination (\puteqn{totzet2}) leads to the total coefficients on ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$,
$$
A^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}+}_{a,n/2}(p)=A^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}}_{a,n/2}(p)+A^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}}_{a,n/2}(p-1)
\eql{comb}$$
and similarly for relative conditions. (\puteqn{comb}) can be inverted as in
(\puteqn{inver}).

In general $\zeta^{\cal M}}\def\mwilde{{\widetilde{\cal M}}_p(s)$ has a pole at $s=0$ which translates into the
$\log\tau$ term in the expansion (\puteqn{expans}), the coefficient being
$$
A'_{\cal M}}\def\mwilde{{\widetilde{\cal M}}(p)=-{\rm Res\,}\zeta^{\cal M}}\def\mwilde{{\widetilde{\cal M}}_p(0).
\eql{logcoeff}$$

These are the only general equations that are needed but the algebra can be
checked by confirming Hodge duality on ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$, in the coefficient form
$$
A^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}+}_{r,n/2}(d+1-p)=A^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}+}_{a,n/2}(p),
\eql{coeffhd}$$
using the easily verified formula,
$$
A^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}\pm}_{n/2}(d-p)=\pm A^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}\pm}_{n/2}(p)
$$
where, the coefficients $A^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}\pm}_{n/2}(p)$ are those resulting from the
combinations of the coexact and `exact' $\zeta$--functions\ on ${{\cal N}}\def\nwilde{{\widetilde{\cal N}}}$ discussed above,
$$
\zeta^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}\pm}_{p}(s)=\zeta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_p(s)\pm\zeta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_{p-1}(s)
\eql{totzet3}$$

\section{\bf 4. The ball and sphere $\zeta$--functions.}

We now consider forms on that cone whose base is
a unit $d$-sphere, $d=D-1$, {\it i.e. } on the $D$-ball. This will be our main
application of the general formalism exposed in
the previous sections. As we have seen, it is sufficient to
look at coexact forms ({\it i.e. } transverse, antisymmetric fields).

The form $\zeta$--functions\ (\puteqn{modcoexzet}) on the $d$-sphere are needed in order to
find the corresponding
heat-kernel coefficients for substitution into the fundamental equation,
(\puteqn{comancoeff}). The spectral properties have been known
for some time [\putref{GandM,BandM,IandK,IandT,CandT,RandT}] and
the coexact $p$-form eigenvalues of the de Rham Laplacian are readily
established to be
$$\mu(p,l)= \big(l+(d-1)/2\big)^2-\big((d-1)/2-p\big)^2,\quad l=1,2,\ldots.
\eql{eigvals}$$

As anticipated, we again witness the important simplification of the Bessel
function order, (\puteqn{reln}), to the $p$-independent form
$$
\nu(p,l)= l+(d-1)/2\,>(d-1)/2
\eql{nu}$$
exactly as in the scalar case. The first consequence is that the absolute
$\zeta$--function\ (\puteqn{coexze}) simplifies to
$$
\zeta^{\rm ball}_{a,p}(s)=\!\!\sum\!\int_\gamma{dk\over2\pi i}k^{-2s}{\partial\over\partial k}\!
\bigg(\tilde d(p)\big(r^{u_a(p)}\ln J_\nu(kr)\big)'\bigg|_{r=1}
+d(p-1)\ln J_\nu(k)\bigg) .
\eql{ballzet}$$

For the time being let us work at a fixed $p$. The sphere coexact $p$-form
degeneracy is
$$
d(p,l)={(2l+d-1)(l+d-1)!\over p!(d-p-1)!(l-1)!(l+p)(l+d-p-1)}.
\eql{degen}$$
We note the symmetry, $d(p,l)=d(d-1-p,l)$ and that $d(d,l)=0$.
In addition, there is a zero mode for $p=0$ and one for $p=d$.

Rewrite (\puteqn{degen}) as
$$
d(p,l)={(l+d-1)!\over p!(d-p-1)!(l-1)!}\bigg({1\over l+p}+{1\over l+d-p-1}
\bigg),
\eql{degen2}$$
and consider, firstly, the sum
$$
\sum_{l=1}^\infty{(l+d-1)!\over(l-1)!}{r^{l+(d-1)/2}\over l+p}
\eql{sum2}$$
with $r=\exp(-\tau)<1$. The idea is that this gives the
square-root heat-kernel, and the sphere $\zeta$--function, (\puteqn{modcoexzet}), follows by
Mellin transform on $\tau$ as in our earlier works, [\putref{Dow1,ChandD}].

The generating function for a given form order can be rewritten using
the identity
$$
\sum_{l=1}^\infty{(l+d-1)!\over(l-1)!}{r^l\over l+p}=
(d-p-1)!\sum_{m=p+1}^d{(m-1)!\over(m-p-1)!}{r\over(1-r)^m}
\eql{iden}$$
which follows easily from recursion.\marginnote{Checked SUMMS.MTH}

There is still an overall factor of $r^{(d-1)/2}$ in (\puteqn{sum2}) and
performing the Mellin transform  produces a series of
Barnes $\zeta$--functions, giving, after the addition of the $p\to d-p-1$ term, the
{\it modified} coexact $\zeta$--function, (\puteqn{modcoexzet}), on the sphere as
$$\eqalign{
\zeta_p^{S^d}(s)=&
\sum_{m=p+1}^d\comb{m-1}p\zeta_{\cal B}
\big(2s,(d+1)/2\mid{\bf1}_m\big)\cr
&+\sum_{m=d-p}^d\comb{m-1}{d-p-1}\zeta_{\cal B}
\big(2s,(d+1)/2\mid{\bf1}_m\big)\cr}
\eql{spcoze}$$
for $0\le p<d$. Obviously $\zeta_d^{S^d}(s)=0$.

(\puteqn{spcoze}) is formally much simpler
than expanding the degeneracy to give a series of Hurwitz $\zeta$--functions\ (a series
noted in passing  by Copeland and Toms, [\putref{CandT}] and frequently
used). This will come later.

When $p=0$, (\puteqn{spcoze}), with the zero mode included according
to (\puteqn{zetzeroin}), gives the known scalar expression.
The first sum reduces to a term
that is cancelled by the zero mode and to a single Barnes $\zeta$--function, the result
being, [\putref{ChandD}], [\putref{BKD}] eqn. (5.7),
$$
\tilde\zeta_0^{S^d}=\zeta_{{\cal B}}( 2s; (d+1)/2\mid {\bf 1}_d)
+\zeta_{{\cal B}}( 2s; (d-1)/2\mid {\bf1}_d ).
\eql{0zeta}$$

This is elegantly seen from a rearrangement of (\puteqn{spcoze}), detailed
in Appendix A, that yields $\zeta_p^{S^d}(s)$ as a finite series of scalar
$\zeta$--functions\ by means of a recursion relating a $p$-form in $d$ dimensions to
a $(p-1)$-form in $d-2$. The series is ($0\le p<(d-1)/2$)
$$\eqalign{
\tilde\zeta_p^{S^d} (s) =\sum_{j=0}^p &(-1)^j \left( d-1-2j \atop p-j
\right)
\tilde\zeta_0^{S^{d-2j}}(s)\cr
&+\delta_{p,0}\bigg({d-1\over2}\bigg)^{-2s}
-(-1)^p\bigg({d-2p-1\over2}\bigg)^{-2s}\cr}
\eql{finser}
$$
and this is one of our basic equations. Duality, (\puteqn{zedual}), can be
used to extend the range of $p$.

When $d$ is odd we have from (\puteqn{0zeta}) the special values for
$q\in\mbox{\open\char90}$,
$$
\tilde\zeta_0^{S^d}(-q)=0,\quad q\ge0,
\eql{qval0}$$
which corresponds to the fact that for scalars the relevant
operator is the (improved) geometrical Laplacian.

From (\puteqn{qval0}) follow the special values of the modified {\it coexact}
function \marginnote{GENBERN.MTH} ($0<p<d$)
$$
\zeta_p^{S^d}(-q)=\tilde\zeta_p^{S^d}(-q)=
(-1)^{p+1}\big((d-1)/2-p\big)^{2q},
\eql{spval}$$
In particular
$$
\zeta_p^{S^d}(0)=(-1)^{p+1},
\eql{0val}$$
and
$$
\zeta_{(d-1)/2}^{S^d}(-q)=0,\quad q>0.
\eql{qval}$$

Hodge duality, as expressed in (\puteqn{zedual}), can be seen explicitly from
(\puteqn{spcoze}). Inclusion of the zero modes obviously violates this duality.

It is also worth pointing out that the + combination of coexact and
exact $\zeta$--functions\ in (\puteqn{totzet3}) on the sphere, in contrast to
(\puteqn{totzet2})
on the ball, does not correspond to the $\zeta$--function\ for any Laplacian.

Finally, we note that $\zeta_p^{S^d}(s)$ does not have a pole at
$s=-1/2$, which is important during the construction of the heat-kernel
coefficients and the functional determinants on ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$. It means that the
$\log$ term is absent in the expansion (\puteqn{expans}), {\it cf } [\putref{BKD}].
No further discussion of this term is given here. Some relevant comments
can be found in the recent works by Bytsenko {\it et al}
[\putref{BCZ1,BCZ2}].

\section{\bf 5. Residues and sphere heat-kernel coefficients.}
To apply (\puteqn{comancoeff}) to the ball it is necessary to know the
coefficients on the sphere. We use (\puteqn{coefff}) on ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$ with the
appropriate $\zeta$--function, in this case either $\zeta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}$ or $\tilde\zeta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}$.

From (\puteqn{finser}), the residue of the modified {\it coexact} sphere
$\zeta$--function\ at $s=k/2$, $k\in\mbox{\open\char90}$, is
$$
{\rm Res\,}\zeta_p^{S^d}(k/2)={1\over(k-2)!}\sum_{j=0}^p(-1)^j
{2^{k-d+2j}D^{(d-2j-1)}_{d-2j-k}
\over (d-2j-1)(d-2j-k)!}\comb{d-2j-1}{p-j}
\eql{resi2}$$
where we have been able to use the results of our earlier work
[\putref{BKD}] for the residues of 0-form $\zeta$--functions. The $D^{(n)}_\nu$ are easily
evaluated generalised Bernoulli polynomials.

The corresponding heat-kernel coefficients are, for ${\rm min}\,([n/2],p,d-p)=
[n/2]$ with $d-k=n$,
$$
{(4\pi)^{d/2}\over|S^d|}A^{S^d}_{n/2}(p)=
(k-1){\Gamma\big((d+1)/2\big)\over\Gamma\big((k+1)/2\big)}
\sum_{j=0}^{[n/2]}{(-1)^j2^{2j}
D^{(d-2j-1)}_{n-2j}\over(d-2j-1)(n-2j)!}\comb{d-2j-1}{p-j}.
\eql{co5}$$
$A^{S^d}_{n/2}(p)$ vanishes when $n$ is odd and less than $d$ because
$D^{(n)}_\nu$ is zero for odd $\nu$.

A useful check is provided by the modified coexact coefficients
$A^{S^d}_{q+d/2}=(-1)^q\zeta_p^{S^d}(-q)/q!$, $q\in \mbox{\open\char90}^+$, for odd $d$.
According to (\puteqn{spval}) these half-odd coefficients are non-zero, in
agreement with the results of [\putref{ELV1}] and [\putref{CandH2}] for the de
Rham coexact $\zeta$--function.
For this case, the only half-odd coefficient is the $q=0$ one,
corresponding
to $\zeta(0)$, and we know that the modified and de Rham coexact heat-kernels
are related by a factor,
$$
K_p^{\rm mod}(\tau)=e^{-\big((d-1)/2-p\big)^2\tau}\,K_p^{\rm de Rham}(\tau),
\eql{hkreln}$$
the expansion of which reproduces exactly (\puteqn{spval}). There are no
half-odd coefficients for $n<d$, nor for $p=0$ when the zero mode is
included.

Similar statements hold for the `total' coefficient, $A^{S^d+}_{n/2}$.
Because of the construction, (\puteqn{totzet3}), of $\zeta_p^{S^d+}$, one
should not, of course, expect a standard geometric form, except when $p=0$.

As a curiosity, constructing the `total' sphere coefficient
({\it cf } (\puteqn{totzet3})) by just adding equation (\puteqn{co5}) for $p$ and
$p-1$, we have
$$
{(4\pi)^{d/2}\over|S^d|}A_{n/2}^{S^d+} =
(k-1){\Gamma\big((d+1)/2\big)\over\Gamma\big((k+1)/2\big)}
\sum_{j=0}^{[n/2]} (-1)^j
\frac{2^{2j-n} D_{n-2j}^{(d-2j-1)}}
{(d-2j-1) (n-2j)!}\left( d-2j\atop p-j \right),
\eql{sphere1}
$$
similar to the general form on the ball developed in the
next section and also in agreement with the results of G\"unther and
Schimming [\putref{GandS}] and Gilkey, {\it e.g. } [\putref{Gilkey1}].

Equation (\puteqn{hkreln}) can be used, in familiar fashion, to give general
formulae for the standard,{\it i.e. } de Rham, coexact heat-kernel coefficients on
the sphere, denoted $B_{n/2}$, somewhat simpler than those derived in
[\putref{CandH2}] and [\putref{ELV1}]. Thus
$$\eqalign{
B^{S^d}_M(p)&=\sum_{N=0}^M {w(p)^{M-N}\over(M-N)!}\,A_N^{S^d}(p)\cr
B^{S^d}_{M+1/2}(p)&=\sum_{N=0}^M {w(p)^{M-N}\over(M-N)!}\,
A_{N+1/2}^{S^d}(p)\cr}
\eql{deRcoeffs}$$
where $w(p)=\big((d-1)/2-p\big)^2$ and $A_{n/2}^{S^d}(p)$ is given by
(\puteqn{co5}). For odd $d$, the only non-zero half-odd coefficient is
$B^{S^d}_{d/2}(p)$, as has already been remarked.

Elizalde {\it et al} [\putref{ELV1}] give selected numerical values for the
total $B$ coefficients obtained from a general formula that involves a
series of Hurwitz $\zeta$--functions\ and is slightly less convenient than the above.
We have found agreement with their values.

The connection between the two expressions is simply that the result of
[\putref{ELV1}] follows from ours if the generalised Bernoulli polynomials
are expanded in ordinary ones. From a numerical point of
view this is unnecessary because there are efficient algorithms that
give the generalised functions directly.

\section{\bf 6. General form of coefficients on the ball.}
We now come to the central part of this paper, the explicit construction
of the heat-kernel coefficients on the ball.

Firstly we can think of checking against the limited results of
[\putref{BBG}] for the coefficients for the form--valued
Laplacian on manifolds with boundary.

From Theorem 1.2 of [\putref{BBG}] applied to the $D$-ball,
the {\it total} coefficient expressions for absolute boundary conditions
are,
(the $+$ superscript is dropped in this section)
$$\eqalign{
{(4\pi)^{d/2}\over|{\rm S}^d|}A^{(D)}_0(p)&={1\over2\sqrt\pi}{1\over(d+1)}
\comb Dp\cr
{(4\pi)^{d/2}\over|{\rm S}^d|}A^{(D)}_{1/2}(p)&=
{1\over4}\bigg[\comb Dp-2\comb {D-1}{p-1}\bigg]\cr
{(4\pi)^{d/2}\over|{\rm S}^d|}A^{(D)}_{1}(p)&=
{1\over\sqrt\pi}{d\over6}\bigg[\comb Dp-6\comb {D-1}{p-1}
+6\comb{D-2}{p-2}\bigg]\cr
{(4\pi)^{d/2}\over|{\rm S}^d|}A^{(D)}_{3/2}(p)&=
{d\over384}\bigg[(13d+2)\comb Dp-4(29d-26)\comb {D-1}{p-1}+\cr
&\hspace{******}288(d-1)\comb{D-2}{p-2}-192(d-1)\comb{D-3}
{p-3}\bigg].\cr}
\eql{BBG12}$$

Assuming, according to our general philosophy, that
$d$ is sufficiently large,
it has been checked that the first three coefficients follow from
our formulation using (\puteqn{comancoeff}), (\puteqn{co5}) and various
binomial identities including the recursion
$$
\comb ab=\comb {a-1}b+\comb{a-1}{b-1}.
\eql{brecurs}$$
The fourth coefficient will be confirmed later by a
more powerful technique, and the series continued.

Although these results were derived on the basis of a restriction  on
$p$, no such restriction exists for the expressions
(\puteqn{BBG12}) in [\putref{BBG}]. The conclusion is that, once the
coefficients are
displayed as explicit functions of $d$ and $p$, they can be continued
outside
the restrictions, the reason being that their particular structure is
guaranteed by the geometric formulation.

These particular checks show that our general formulae are correct
but it is not feasible to pursue the further evaluation of the coefficients
in this way by algebraic rearrangement of the produced expressions, since
it involves awkwardly repeated application of the binomial recursion and of
other identities.
A systematic approach to the evaluation of any coefficient is better provided
by fitting unknowns in a general form, and this process will now be set in
train.

From basic theory ({\it e.g. } [\putref{Gilkey1}]) the geometric expression on a
flat,
bounded $D$-manifold, ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$, is, up to terms involving derivatives of the
extrinsic curvature $\bka$,
$$
c(n)(4\pi)^{d/2}A^D_{n/2}(p)=\int_{\partial{\cal M}}\def\mwilde{{\widetilde{\cal M}}}b_{\bf n}(D,p)\sum_{\bf n}
\big({\rm tr\,}(\bka^{n_1}){\rm tr\,}(\bka^{n_2})\ldots\big)
\eql{flgen}$$
with
$$\eqalign{
c(n)&=2\sqrt\pi,\quad n\,\, {\rm even}\cr
&=1,\quad n\,\,{\rm odd}\cr
&=2(d+1)\sqrt\pi,\quad n=0\cr}
$$
and where ${\bf n}=(n_1,n_2,\ldots)$ is a partition of $n-1$. For
convenience
the $n=0$ term has been included although it is really a volume
contribution.
We seek the $b_{\bf n}(D,p)$.

For the $D$-ball, (\puteqn{flgen}) reduces to
$$
c(n)(4\pi)^{d/2}A^D_{n/2}(p)=|{\rm S}^d|\,\sum_{\bf n}d^{|{\bf n}|}
b_{\bf n}(D,p)=|{\rm S}^d|\,\sum_{k=1}^{n-1}d^k\,b_k^{(n)}(D,p)
\eql{ballg}$$
where $|{\bf n}|$ is the number of components in the partition and
$b_k^{(n)}$,
is the sum of those $b_{\bf n}$ for which $|{\bf n}|=k$,
$$
b_k^{(n)}(D,p)=\sum_{\bf n}b_{\bf n}(D,p)\bigg|_{|{\bf n}|=k}.
$$
These are the only combinations that can be determined from working on
the ball.

The numerical multipliers $b_{\bf n}(D,p)$ satisfy the binomial recursion
$$
b_{\bf n}(D,p)=b_{\bf n}(D-1,p)+b_{\bf n}(D-1,p-1),
\eql{bbrec} $$
proved by crossing ${\cal M}}\def\mwilde{{\widetilde{\cal M}}$ with a unit circle, [\putref{BBG}].
This relation is what has become of the more familiar statement of
dimension-independence for scalars.

Thus $b_{\bf n}(D,p)$ can be expanded as a linear combination of binomial
coefficients $\comb{D+a}{p+b}$ for varying $a$ and $b$. Since
$b_{\bf n}(D,p)$ vanishes for $p$ outside the range $0$ to $D$,
$a$ must equal $b$ and be nonpositive. Consequently the expansion reads
$$
b_{\bf n}(D,p)=\sum_{m=0}^nM_{{\bf n},m}\comb{D-m}{p-m}
\eql{bexp}$$
which is the boundary version of the G\"unther and Schimming form
[\putref{GandS}].
The limits have been fixed by noting that they must be independent of
both $D$ and $p$ and, therefore, can be set by considering the particular
value $D=n$, for which there are $n+1$ independent constants implying $n+1$
terms in the sum. The trivial first coefficient, $b_0=\comb D p$, can
also be used to pin down the limits to those shown.
The conclusion is that, on the ball, the general form can be written,
$$
{c(n)(4\pi)^{d/2}\over|{\rm S}^d|}A^{(D)}_{n/2}(p)=
\sum_{m=0}^nP_m^{(n)}(d)\comb{D-m}{p-m}
\eql{genformcoeff}$$where
$P_m^{(n)}(d)$ is a polynomial of degree $n-1$ in $D$.
For $n>1$
$$
P_m^{(n)}(d)=\sum_{\bf n}M_{{\bf n},m}d^{|{\bf n}|}
=\sum_{k=1}^{n-1} M^{(n)}_{k,m}\,d^k
\eql{ppoly}$$where the $M^{(n)}_{k,m}$ are constants.

Our method is simply that, for fixed $n$, knowing the left-hand side
for $0\le p\le n$, (\puteqn{genformcoeff}) can be inverted for the
$P_m^{(n)}(d)$, keeping $d$ unspecified. The method employed in
[\putref{BBG}] uses the particular values $b_{\bf n}(n,p)$ to invert
(\puteqn{bexp}). Our technique is more flexible in practice, although the
analysis in [\putref{BBG}] does bring out clearly the amusing fact that
the general coefficient
can be generated from just the values $A^{(n)}_{n/2}(p)$, $0\le p\le n$.

In detail, we firstly note that, for $p=0$ to $n$, the matrix of binomial
coefficients on the right-hand side of (\puteqn{genformcoeff}) is triangular
so that the inversion follows recursively by forward substitution,
$$
P_m^{(n)}(d)={c(n)(4\pi)^{d/2}\over|{\rm S}^d|}A_{n/2}^{(D)}(m)-
\sum_{\mu=0}^{m-1}\comb {D-\mu}{m-\mu}P^{(n)}_\mu(d),\quad m=0,\ldots, n.
\eql{fsub}$$

The driving coefficients $A_{n/2}^{(D)}(m)$ ($0\le m\le n$) will be
determined from (\puteqn{comancoeff}) as polynomials in $d$ because,
for given numerical values of $p$ and $n$, the sphere coefficients,
(\puteqn{co5}), are obviously such polynomials.

It should be stated that the $\zeta$--function\ on the ball is a standard one, and that
the heat-kernel coefficients are geometric (leading to the general form
(\puteqn{genformcoeff})). In this case it is quite permissible to evaluate the
polynomials in whatever parameter region is convenient and then continue,
despite the fact that they involve coefficients on the sphere for which
this might not be possible.

Evaluation is a routine machine matter, the first outcome
of which is $A^{(D)}_{3/2}(p)$ in (\puteqn{BBG12}). Some results are
given below in the form of matrices of the constants $M^{(n)}_{k,m}$ in
(\puteqn{ppoly}), \marginnote{MMATR.MTH}
$${\bf M}_a^{(3)}=
\left(\matrix{{2\over192}&{13\over48}&-{3\over4}&-{1\over2}\cr
{13\over384}&-{29\over96}&{3\over4}&1\over2\cr}\right)
\eql{al3}
$$
\vskip15truept
$${\bf M}_a^{(4)}=
\left(\matrix{{4\over{135}}&-{{164}\over{315}}&{{16}\over5}&-{{16}\over 3}&
{8\over3}\cr
{1\over{45}}&{{92}\over {105}}&-{{74}\over {15}}& {\scriptstyle8}&
{\scriptstyle-4}\cr
{1\over{27}}&-{{136}\over{315}}&{{26}\over{15}}&-{8\over3}&{4\over 3}\cr}
\right)
\eql{al4}$$
\vskip15truept
$${\bf M}_a^{(5)}=
\left(\matrix{
{{77}\over {15360}}&
{{77}\over {960}}&
 -{{191}\over {192}}&
{{19}\over 6}&
 -{{15}\over 4}&
{3\over 2}\cr
{{235}\over {36864}}&
-{{263}\over {1440}}&
{{1475}\over {768}}&
 -{{47}\over 8}&
{{55}\over 8}&
-{{11}\over 4}\cr
{{139}\over {61440}}&
{{1987}\over {15360}}&
-{{1769}\over {1536}}&
{{157}\over {48}}&
-{{15}\over 4}&
{3\over 2}\cr
{{2041}\over {737280}}&
-{{3787}\over {92160}}&
{{347}\over {1536}}&
-{9\over {16}}&
 {5\over 8}&
 -{1\over 4}\cr}\right).
\eql{al5}$$
\vskip15truept

A subscript has been added to indicate that these values are for absolute
boundary conditions.
Although any relative coefficient can be found from the absolute ones
using duality, in order to obtain the general form
the entire analysis should be repeated. A tactically better way combines
these procedures and consists
of using the recursive nature of the relative coefficients to write
down a general form, precisely as in (\puteqn{genformcoeff}), together with the
corresponding solution, (\puteqn{fsub}), but now where the values of the
coefficients for $p=0$ to $n$ are determined by duality in terms
of already evaluated absolute quantities.  In any case, one rapidly finds
that the first coefficient is the same, that the second is reversed in sign
and that the remaining ones (up to $n=5$) are contained in the matrices,
$$
{\bf M}_r^{(3)}=\left(\matrix{
{5\over {192}}&
-{13\over {48}}&
{3\over 4}&
-{1\over 2}\cr
-{7\over {384}}&
{{29}\over {96}}&
 -{3\over 4}&
{1\over 2}\cr}
\right)
$$

$$
{\bf M}_r^{(4)}=\left(\matrix{
{8\over {189}}&
-{{172}\over {315}}&
{{16}\over 5}&
-{{16}\over 3}&
{8\over 3}\cr
-{{11}\over {315}}&
{{104}\over {105}}&
-{{74}\over {15}}&
{\scriptstyle8}&
{-\scriptstyle4}\cr
{1\over {189}}&
-{{116}\over {315}}&
{{26}\over {15}}&
-{8\over 3}&
{4\over 3}\cr}
\right)
$$
\vskip10truept
$$
{\bf M}^{(5)}_r=\left(\matrix{
{{109}\over {15360}}&
-{{29}\over {320}}&
{{193}\over {192}}&
-{{19}\over 6}&
{{15}\over 4}&
-{3\over 2}\cr
-{{1049}\over {184320}}&
{{1247}\over {5760}}&
-{{1501}\over {768}}&
{{47}\over 8}&
-{{55}\over 8}&
{{11}\over 4}\cr
{{47}\over {61440}}&
-{{709}\over {5120}}&
{{1783}\over {1536}}&
-{{157}\over {48}}&
{{15}\over 4}&
-{3\over 2}\cr
{{13}\over {147456}}&
{{2467}\over {92160}}&
-{{325}\over {1536}}&
{9\over {16}}&
-{5\over 8}&
{1\over 4}\cr}
\right).
\eql{rem5}$$
\vskip10truept
\noindent It is a matter of a few minutes by machine algebra to calculate
larger matrices.

Knowledge of these constants enables restrictions to be placed on the
multipliers, $b_{\bf n}(D,p)$, in the general geometric form of the
coefficients, (\puteqn{flgen}).

Denoting the set of $b_k^{(n)}(D,p)$, $(1\le k\le n-1)$,
by the vector ${\bf b}^{(n)}$, the vector of binomial coefficients
$\comb {D-m}{p-m}$, $(0\le m\le n)$, by ${\bf c}^{(n)}$ and the set of
powers $d^k$ ($1\le k\le n-1$) by $\bf d^{(n)}$, equation
(\puteqn{genformcoeff}) takes the matrix form
$$
{c(n)(4\pi)^{d/2}\over|{\rm S}^d|}A^{(D)}_{n/2}(p)=\widetilde{{\bf d}^{(n)}}
{\bf M}^{(n)}{\bf c}^{(n)}
\eql{coefmat}$$
and the restrictions are
$$
{\bf b}^{(n)}={\bf M}^{(n)}{\bf c}^{(n)}.
\eql{rest}$$

For example, for $n=5$ there are 4 partitions so
$$
b_1^{(5)}=b_{4},\quad
b_2^{(5)}=b_{1,3}+b_{2^2},\quad
b_3^{(5)}=b_{1^2,2},\quad
b_4^{(5)}=b_{1^4}
$$
and only these particular combinations are known using (\puteqn{rest}) and
(\puteqn{al5}) or (\puteqn{rem5}).
It is clear that one can always unambiguously determine $b^{(n)}_{n-1}$ and
$b^{(n)}_{1^{n-1}}$.

We finally note that the $n=4$ values at $D=4$ agree with those in Table 2
of Moss and Poletti [\putref{MandP}].\marginnote{MMATR.MTH}
\section{\it Generating functions.}
A generating function approach to the form order, $p$, could be introduced
at this coefficient stage by defining
$$
\overline A_{n/2}^{(D)}(\sigma)=
\sum_{p=0}^D(-\sigma)^p A^{(D)}_{n/2}(p)
={|{\rm S}^d|\over c(n)(4\pi)^{d/2}}\sum_{m=0}^n(-\sigma)^m(1-\sigma)^{D-m}P_m^{(n)}
(d).
\eql{genfcoeff}$$

In particular the value $\sigma=-1$ is required ({\it cf } [\putref{BBG}] Thm. 1.3),
$$
\overline A_{n/2}^{(D)}(-1)
={4\sqrt\pi\over c(n)\Gamma\big((d-1)/2\big)}\sum_{m=0}^n2^{-m}P_m^{(n)}(d)
\eql{sumcoef}$$
which is a convenient way of organising and computing this quantity, at
this point.

There is topological interest in the case $\sigma=1$ and we see
from (\puteqn{genfcoeff}) that
$$
\overline A_{a,n/2}^{(D)}(1)=0,\quad n<D;\quad
\overline A_{a,D/2}^{(D)}(1)={(-1)^D2^{1-d}\sqrt\pi
\over c(D)\Gamma\big((d-1)/2\big)}P^{(D)}_D(d)=\chi({\cal M}}\def\mwilde{{\widetilde{\cal M}})=1
\eql{euler}$$
which provides a check of the Gauss-Bonnet theorem. If relative
coefficients are used we find $\chi({\cal M}}\def\mwilde{{\widetilde{\cal M}},\partial{\cal M}}\def\mwilde{{\widetilde{\cal M}})=-1$, which is correct.

\section{\bf 7. Value at zero.}
Special interest, both mathematical and physical, attaches itself to the
particular value
$\zeta^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}+}_p(0)$ which merits individual treatment, {\it e.g. }
[\putref{MandP, Esposito, Barv}]. Further,
as already said, these values play an
important role in the algebraic computations of Bla\v zi\'k {\it et al}
[\putref{BBG}].

Generally, for absolute conditions, the coexact expression is
$$\eqalign{
\zeta^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}}_p&(0)=-{1\over2}\big(\tilde\zeta^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_{p}(-1/2)
+\zeta^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_{p-1}(-1/2)\big)+{1\over4}\big(\tilde\zeta^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_{p}(0)-
\zeta^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_{p-1}(0)\big)\cr
&-\sum_{k=1}^d{1\over k}\bigg(\zeta_R(-k)\big({\rm Res\,}\zeta^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_{p}(k/2)
+{\rm Res\,}\zeta^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_{p-1}(k/2)\big)
-\big(-u_a(p)\big)^k{\rm Res\,}\zeta^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_{p}(k/2)\bigg).\cr}
\eql{0val2}$$

For ${\cal N}}\def\nwilde{{\widetilde{\cal N}}={\rm S}^d$ the residues are contained in (\puteqn{resi2}) and the
values of the $\zeta$--function\ found from (\puteqn{spcoze}) or (\puteqn{finser})
(there are no zero modes on the sphere {\it in the sense that} $\nu>0$).

An alternative evaluation consists of using the standard formula
$$
\zeta_p^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}+}(0)=A^{(D)}_{D/2}(p)-\beta^{\cal M}}\def\mwilde{{\widetilde{\cal M}}_p
\eql{standeq}$$
together with an explicitly computed general coefficient
(\puteqn{genformcoeff}).
The absolute Betti numbers of the ball are $\beta_0=1$, the rest vanishing.

The direct evaluation via (\puteqn{0val2}) is the most efficient, numerically.
Machine computation shows agreement between these two approaches and also
with the values presented in [\putref{ELV2}], except that the $p=0$ numbers
for $d=6$ and $7$ differ.

It is a remarkable fact that in using (\puteqn{comancoeff}), leading to
(\puteqn{genformcoeff}) and thence to (\puteqn{standeq}), one does not have to worry
about the `zero modes' in say (\puteqn{zetzeroin}), whilst such contributions
are vital when evaluating directly from (\puteqn{0val2}).

Finally, using the inverse (\puteqn{inver}) and noting that $\zeta^{\cal M}}\def\mwilde{{\widetilde{\cal M}}_D(0)$
vanishes, we reconfirm the topological formula
$
\chi=\sum_{p=0}^D(-1)^p\,\beta^{\cal M}}\def\mwilde{{\widetilde{\cal M}}_p=\sum_{p=0}^D(-1)^p\, A^D_{D/2}(p).
$
\section{\bf 8. Form functional determinants.}

In this section we consider functional determinants for $p$-forms. As
we have seen in the calculation of the heat-kernel coefficients, it will be
enough to consider the determinant associated with the coexact $\zeta$--function\
(\puteqn{coexze}), this being simply a combination of Robin and
Dirichlet contributions. It is immediately appreciated that the results
derived in [\putref{BKD}], eq.~(9.8) for Dirichlet boundary conditions
and  eq.~(11.2) for Robin conditions, remain valid once the base $\zeta$--function\
there is replaced with the $p$-form base $\zeta$--function,
eq.~(\puteqn{modcoexzet}), with, in addition,
$$
\zeta^{{\cal N}+1}_p (s) = \sum_{n=1}^{\infty}\sum
\frac{d(p)}{\big(\nu (p) +n\big)^{s}},\,\,\,\,{\rm and}\,\,\,\,\,
\zeta^{{\cal N}}_p (s,u) = \sum
\frac{d(p)}{\big(\nu (p) +u\big)^{s}},
\eql{zets}$$
and the related quantities, ${\tilde \zeta} _p ^{{\cal N} +1} (s)$,
${\tilde \zeta} _p ^{{\cal N}} (s,u)$, if the zero modes are included as
in (\puteqn{zetzeroin}).

The coexact determinant for $p\geq 1$
combines Robin with Dirichlet and is determined, for absolute conditions, by
the derivatives,
$$\eqalign{
{\zeta'}^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}}_p(0)=&{\tilde \zeta}^{\prime{\cal N}}\def\nwilde{{\widetilde{\cal N}}+1}_p(0)
+{\zeta '}^{{\cal N}}\def\nwilde{{\widetilde{\cal N}} +1}_{p-1}(0)+
{\tilde \zeta}^{\prime{\cal N}}\def\nwilde{{\widetilde{\cal N}}} _p\big(0,u_a (p)\big)
+\ln 2\bigg({\tilde \zeta} ^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_p (-1/2)
+\zeta^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_{p-1}(-1/2)\cr
&+2\sum_{i=1\atop i\,\,odd}^d{\rm Res\,}\zeta_p^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}
(i/2)\, M_i\big(1,u_a(p)\big)+2\sum_{i=1}^d{\rm Res\,}\zeta_{p-1}^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}} (i/2)\,
D_i(1)\bigg)\cr
&+2\sum_{i=1 \atop i\,\, odd}^d {\rm Res\,}\zeta_p^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}} (i/2)
\bigg( M_i\big(1,u_a(p)\big)\sum_{k=1}^{i-1} 1/k \cr
&\hspace{*********}+\int_0^1 dt\,\,
\frac{M_i\big(t,u_a (p)\big)-tM_i\big(1,u_a(p)\big)}{t(1-t^2)}\bigg)\cr
&+2\sum_{i=1 \atop i\,\, even}^d {\rm Res\,}\zeta_p^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}} (i/2)
\bigg( M_i\big(1,u_a(p)\big) \sum_{k=1}^{i-1} 1/k \cr
&\hspace{*********}+\int_0^1 dt\,\,
\frac{M_i\big(t,u_a (p)\big)-t^2 M_i\big(1,u_a(p)\big)}{t(1-t^2)}\bigg)\cr
&+2\sum_{i=1}^d {\rm Res\,}\zeta_{p-1}^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}} (i/2)
\bigg(D_i (1)\sum_{k=1}^{i-1} 1/k +\int_0^1 dt\,\,
\frac{D_i (t)-tD_i (1)}{t (1-t^2)}\bigg).}
\eql{det}$$
The $M_i$ and $D_i$ are known polynomials associated with the asymptotic
expansion of Bessel functions and, to avoid repetition, we refer to
[\putref{BKD}], and to references therein, for more details. Some information
can be found incidentally in Appendix D.

For $p=0$ a small extra consideration is necessary. As we have mentioned,
absolute 0-forms are Neumann scalars
and the Robin parameter is then $u_a (0) = -(d-1)/2$. Looking
at $\zeta_p ^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}} (s,u)$ in eq.~(\puteqn{zets}) it is seen that, for
$u=-\nu (p)$, one encounters a branch cut occasioned by (incorrectly)
including the true zero mode for Neumann conditions. The technical
reason is that the asymptotic expansion for these
specific Robin parameters is slightly different from the others.

The easiest way of taking this into account is to subtract the contribution
in (\puteqn{zets}) for $u\neq -\nu (p)$, then take the limit as $u\to
-\nu (p)$ and finally to add the correct contribution for $u=-\nu (p)$.

For Neumann conditions the end result is
$$\eqalign{
{\zeta'}^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}}_0 (0) =&{\tilde \zeta}^{\prime{\cal N}}\def\nwilde{{\widetilde{\cal N}}+1}_0 (0)
+\ln 2\bigg(\tilde \zeta^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}_0 (-1/2)
+2\sum_{i=1\atop i\,\,odd}^d{\rm Res\,}\zeta_0^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}(i/2)\, M_i\big(1,u_a(0)\big)
\bigg)  \cr
\noalign{\vskip 5truept}
&+2\sum_{i=1 \atop i\,\, odd}^d {\rm Res\,}\zeta_0^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}} (i/2)
\bigg( M_i\big(1,u_a(0)\big)\sum_{k=1}^{i-1} 1/k \cr
\noalign{\vskip 5truept}
&\hspace{*********}+\int_0^1 dt\,\,
\frac{M_i\big(t,u_a (0)\big)-tM_i\big(1,u_a(0)\big)}{t(1-t^2)}\bigg)\cr
\noalign{\vskip 5truept}
&+2\sum_{i=1 \atop i\,\, even}^d {\rm Res\,}\zeta_0^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}} (i/2)
\bigg( M_i\big(1,u_a(0)\big) \sum_{k=1}^{i-1} 1/k \cr
\noalign{\vskip 5truept}
&\hspace{*********}+\int_0^1 dt\,\,
\frac{M_i\big(t,u_a (0)\big)-t^2 M_i\big(1,u_a(0)\big)}{t(1-t^2)}\bigg)\cr
\noalign{\vskip 5truept}
&+\lim_{u\to -(d-1)/2 } \left(
{\tilde \zeta _0} ^{\prime{\cal N}}\def\nwilde{{\widetilde{\cal N}}} (0,u) +\ln \left(
(d-1)/2+ u \right)\right)
+\ln \big(d+1\big). \cr}
\eql{detneu}
$$
The relative, Dirichlet expression is given in [\putref{BKD}].

Equations (\puteqn{det}) and (\puteqn{detneu}) are expressions on the generalised
cone and we apply them now to the ball,
for which all quantities have already been treated, apart from
$\zeta^{{\cal N}+1}_p (s)$
($\tilde \zeta^{{\cal N}+1}_p (s)$)
and $\zeta^{{\cal N}}_p (s,u)$
($\tilde \zeta^{{\cal N}}_p (s,u)$).
For these remaining $\zeta$--functions\ one immediately finds, along the lines of
[\putref{BKD}],
$$\eqalign{
\tilde \zeta^{{\cal N}+1}_p (s)=&
\sum_{m=p+1}^d\comb{m-1}p\zeta_{\cal B}
\big(s,(d+3)/2\mid{\bf1}_{m+1}\big)
+\zeta_R\big(s; (d+3)/2)\big)\delta_{pd}
\cr
&+\sum_{m=d-p}^d\comb{m-1}{d-p-1}\zeta_{\cal B}
\big(s,(d+3)/2\mid{\bf1}_{m+1}\big)
+ \zeta_R\big(s; (d+1)/2)\big)\delta_{p0}\cr}
\eql{zetu1}$$
and
$$\eqalign{
\tilde \zeta^{{\cal N}}_p (s,u_a) = &\sum_{m=p+1}^d\comb{m-1}p\zeta_{\cal B}
\big(s,p+1\mid{\bf1}_{m}\big)\cr
&+\sum_{m=d-p}^d\comb{m-1}{d-p-1}\zeta_{\cal B}
\big(s,p+1\mid{\bf1}_{m}\big).\cr
\noalign{\vskip 5truept}
&\hspace{****}+\delta_{p0}\big((d-1)/2+u\big)^{-s}
+\delta_{pd} (d+1)^{-s}.\cr}
\eql{zetu}$$
The contribution of the zero modes is visible and
$ \zeta^{{\cal N}+1}_p (s)$, $ \zeta^{{\cal N}}_p (s,u_a) $ need not
be stated. It is clearly seen that the limit $u\to-(d-1)/2$ in
(\puteqn{detneu}) is well defined because the logarithm is cancelled by the
last term in (\puteqn{zetu}).

Using (\puteqn{zetu1}) and (\puteqn{zetu}) in
(\puteqn{det}) and (\puteqn{detneu}), the determinants emerge as derivatives
of the Barnes $\zeta$--function\ at $s=0$. These may be expanded using Stirling
numbers,
$$\eqalign{
\zeta_{{\cal B}} \big(s,a\mid {\bf1}_m) &= \sum_{n=0}^{\infty}
\comb{m+n-1}m (a+n)^{-s} \cr
\noalign{\vskip 5truept}
&= \sum_{k=1}^{m}{(-1)^{k+m}\over
(k-1)!(m-k)!}B^{(m)}_{m-k}(a)\,\zeta_R (s+1-k,a) \cr}
\eql{deg}$$
to give derivatives of Hurwitz or Riemann $\zeta$--functions, [\putref{Barnesb}] p.433.
The derivatives presented in Appendix B were obtained in this way.
It is easy to find the expression in any dimension $d$ and for
any value of $p$.

The presentation of our results is
a little more explicit than the corresponding ones of
[\putref{ELV2}] in that only derivatives of Riemann $\zeta$--functions\ are
involved, there being no integrals over $\Gamma$ or $\Psi$ functions.
In fact, the integrals in [\putref{ELV2}] can be done,
bringing exact agreement between the two sets of expressions.
This agreement is elaborated, in its essentials, in Appendix C.

It should be noted that, as a consequence of an earlier error in
[\putref{BGKE}] for Neumann 0--forms, a $\log2$ term should be added to the
results in [\putref{ELV2}] for $p=0$ and $d\ge2$.

\section{\bf 9. Spectral properties of spinors.}
For the sake of completeness, the method of [\putref{BKD}] is now employed
to put the results of the earlier work on spinors, [\putref{DABK}], on a
secure and systematic footing. For any undefined quantities and for further
explanations, refer to this paper.
\vskip 10truept
\noindent{\it Spectral boundary conditions.}
\vskip10truept
In [\putref{DABK}] it was shown that spectral boundary
conditions lead to
$$
J_{n+ D/2 -1} (k) =0 ,\qquad n\in \mbox{\open\char90},\eql{1}
$$
with the degeneracy
$$
d(n) =d_S \,\comb{n+D-2}n.\eql{2}
$$
Having in mind the treatment of the scalar field in [\putref{BKD}] we define
the spinor base $\zeta$--function\ appropriate for the ball,
$$
\zn s\right) = \sum_{n=0}^\infty d(n) \left(n+D/2 -1\right)^{-2s}.\eql{3}
$$
This choice guarantees that all formulas in terms
of $\zeta_{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}$ in section 3 of [\putref{BKD}] remain valid. Also the
polynomials $D_i (t)$ defined there are unchanged. In particular, eq.~(4.8)
is still true, the only difference is in the base $\zeta$--function\ used for
the calculation of $A_{n/2}^{{\cal N}}\def\nwilde{{\widetilde{\cal N}}}$, the base heat-kernel coefficients.

Let us continue with the treatment of $\zn s\right)$ for the ball. It has
the especially simple form
$$
\zn s\right) =d_S\, \zeta_{{\cal B}}( 2s, (d-1)/2\mid{\bf 1} _d ).\eql{4}
$$
To obtain the heat-kernel coefficients using eq.~(4.8) of [\putref{BKD}],
we need only the residues in (\puteqn{4}).
Using the known values of the Barnes $\zeta$--function\ in eq.~(4.8) of [\putref{BKD}] one
regains all the results of [\putref{DABK}] for the heat-kernel coefficients
including the conjecture for their general form.

The determinants come out similarly.
Due to the identical structure of the present problem, eq.~(9.8) of
[\putref{BKD}] remains valid once the base zeta
function there is replaced by (\puteqn{3}). The only `unknown' quantity is
$$
\zeta_{{\cal N}}\def\nwilde{{\widetilde{\cal N}} +1} (s) =d_S\, \zeta_{{\cal B}}( s,(d+1)/2 \mid{\bf 1} _{d+1}).\eql{7}
$$
Using (\puteqn{deg}) this can be expanded in Hurwitz-Riemann $\zeta$--functions, and
complete agreement with the results exhibited in [\putref{DABK}] is again
quickly and economically found.

In Appendix D we prove the curious identity for the spectral heat-kernel
coefficients,
$$
A_{(D-2)/2}^{(S)}(D) =0,
\eql{siden1}$$
observed numerically in I, the significance of which has not yet been
discovered.

\section{\it Local boundary conditions.}
For local boundary conditions a few more technical details are necessary.
The eigenvalue condition is, [\putref{DABK}],
$$
J_{n+D/2}^2 (k) -J_{n+D/2 -1} ^2 (k) =0,\quad n\in\mbox{\open\char90}\eql{9}
$$
with the mode degeneracy
$$
d(n) ={d_S\over2}\,\comb{n+D-2}n, \eql{10}
$$
which is half that of the spectral case. Defining the base
$\zeta$--function\ as in (\puteqn{3}) we have
$$
\zn s\right) =\frac 1 2 d_S\, \zeta_{{\cal B}}( 2s,D/2 -1\mid{\bf 1}_d ).\eql{11}
$$
The polynomials describing the asymptotics are a bit different,
however, and read
$$
D_i (t) =\sum_{a=0}^{2i} x_{i,a} t^{i+a},
$$
which leads to a slightly different form of the $A_i (s)$. These are
found to be (one only has to refer to [\putref{KandC}] eq.~(2.18), and
replace the $\zeta$--functions\ by the base $\zeta$--function),
$$\eqalign{
A_{-1} (s) =& \frac 1 {2\sqrt{\pi}} \frac{\Gamma (s-1/2)}{\Gamma (s+1)}
\zn s-1/ 2 \right), \cr
A_0 (s) =& -\frac 1 {2\sqrt{\pi}} \frac{\Gamma (s+1/2)}{\Gamma (s+1)}
\zn s\right),\cr
A_i (s) =& -\frac 1 {\Gamma (s)} \zn s+i/ 2 \right)
\sum_{a=0}^{2i} x_{i,a} \frac{\Gamma \left( s+(i+a)/2\right)}
{\Gamma \left((i+a)/2 \right)}.\cr}
\eql{12}$$
Due to the different $A_0 (s)$ and $A_i (s)$ the heat-kernel
coefficients this time read
$$\eqalign{
A_{n/2}^{{\cal M}}\def\mwilde{{\widetilde{\cal M}}} =& \frac{\Gamma\left((D-n-1)/2 \right)}{\sqrt{\pi}
(D-n)} \Res\,\,\zn (D-n-1)/2\right) \cr
&\hspace{*}-\frac{\Gamma\left((D-n+1)/2 \right)}{\sqrt{\pi} (D-n)}
\Res\,\,\zn (D-n)/2 \right) \cr
&\hspace{***}-\sum_{i=1}^{n-1} \Res\,\,\zn (D-n+i)/2 \right)
\sum_{a=0}^{2i} x_{i,a} \frac{\Gamma \left( (D-n+i+a)/2 \right)}
{\Gamma \left( (i+a)/2\right) }.\cr}
\eql{13}$$

It would be possible to rewrite the right-hand side in terms of the spinor
heat-kernel coeffcients on the base, leading to an equation like
(\puteqn{comancoeff}).

For the ball, using (\puteqn{resi2}) for the residues of $\zn s\right)$,
(\puteqn{11}),
all results of [\putref{DABK}] are immediately reproduced in an organised
and rapid fashion. Let us therefore proceed to the  determinant.

As a result of (\puteqn{12}) one has
$$\eqalign{
A_{-1} ' (0) =& 2\, (\ln 2 -1)\, \zn -1/2 \right) -\zeta_{{\cal N}}\def\nwilde{{\widetilde{\cal N}}} ' \left(
-1/2 \right),  \cr
A_0 ' (0) =& \ln 2 \,\zn 0\right) -\frac 1 2 \zeta_{{\cal N}}\def\nwilde{{\widetilde{\cal N}}} ' (0) ,
\cr
A_i ' (0) =& -\sum_{a=0}^{2i} x_{i,a} \bigg( PP\,\, \zn i/ 2 \right)
+\gamma \Res\,\,\zn i/2 \right) +\cr
&\hspace{**************}\Res\,\,\zn i/ 2 \right) \psi \left(
(a+i)/ 2 \right) \bigg),\cr}
\eql{14}$$
with characteristic differences to the spectral case.
Further contributions come from
$$
{\cal Z} ' (0) = \sum_{n=0}^\infty d(n) {{\cal Z} ^{\nu}} ' (0),\quad
\nu=n+D/2-1,\eql{15}
$$
and
$$
{{\cal Z} ^{\nu}} ' (0) = 2\ln\Gamma (\nu +1) +2\nu -2\nu\ln \nu
-\ln (2\pi\nu ) +\sum_{n=1}^{D-1} \frac{D_n (1)}{\nu^n}. \eql{16}
$$
This follows from the results preceding eq.~(3.1) in [\putref{KandC}].

Formally (\puteqn{16}) is twice the scalar case. It  also shows that
$D_n (1) =2\zeta_R (-n) /n$ and so the contribution ${\cal Z} ' (0)$ is
known from the scalar field calculation. Adding up, everything fits nicely
and
the final answer reads
$$\eqalign{
{\zeta _{1/2}^{lo}}' (0) =& \ln 2 \big( 2\zn -1/ 2 \right) +
\zn 0\right)\big) +2\zeta_{{\cal N}}\def\nwilde{{\widetilde{\cal N}} +1} ' (0)  \cr
&+4\ln 2 \sum_{i=1}^{D-1} \frac{\zeta _R (-i) } i \Res\,\,\zn i/ 2 \right)
\cr
&+2\sum_{i=1}^{D-1} \Res\,\,\zn i/ 2 \right) \left[\frac {2\zeta_R (-i)} i
\sum_{k=1}^{i-1} \frac 1 k
+\int_0^1 dt\,\,\frac{D_i (t) -tD_i (1)}{t(1-t^2)}\right].\cr}
\eql{17}$$
Specialising to the ball, all results of [\putref{DABK}] are thereby
established again in a systematic and general way.

\section{\bf 10. Conclusion.}
Our essential results are, firstly, the expression (\puteqn{comancoeff}) for
the
coexact heat-kernel coefficients on the generalised cone in terms of those
on its base. Secondly, (\puteqn{genformcoeff}) gives
any ball heat-kernel coefficient, with explicitly computable polynomials,
(\puteqn{ppoly}), and leads to the restriction (\puteqn{rest}) on the general
coefficients. Thirdly, we draw attention to the formula (\puteqn{det}) for the
functional determinant and finally to the expression (\puteqn{finser}) for
the modified $p$-form coexact $\zeta$--function\ on the sphere as a finite sum of
(conformal) 0-form functions.

To save excessive analysis, we have concentrated on the simplest generalised
cone, {\it i.e. } the ball. As a continuation, more complicated bases, ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$, can
be
considered. In the earlier paper, [\putref{BKD}], we looked at the case when
the
base was a sphere of non-unit radius, say $a$, calling this a monopole.
Then, in place of (\puteqn{nu}) for the order of the Bessel functions, we have
$$
\nu^2(l,p)={1\over a^2}\big(l+(d-1)/2\big)^2-\big((d-1)/2-p\big)^2
\big({1\over a^2}-1\big)
\eql{nu2}$$
which is not a perfect square unless $p=(d-1)/2$. We therefore expect that
a pole at $s=0$ in $\zeta^{\cal M}}\def\mwilde{{\widetilde{\cal M}}_p(s)$ will arise even for the case of the de Rham
Laplacian. However, unlike the scalar ($p=0$) case, it does not seem
possible to achieve a perfect square, and hence to avoid the pole, simply by
attaching a term to the de Rham Laplacian. This obstruction
is probably related to  the fact that there is no direct way of making the
equation conformally invariant except for the special value $p=(d-1)/2$.
It is possible to use Branson's operator [\putref{Branson}] but we would then
lose contact with the body of work associated with the de Rham Laplacian.

The lack of conformal invariance can be appreciated in the structure of the
coexact $\zeta$--function\ on the sphere, (\puteqn{finser}), which is a combination of
improved 0-form $\zeta$--functions\ each of which is conformal in a different dimension.

Further possible choices for ${\cal N}}\def\nwilde{{\widetilde{\cal N}}$ include hyperbolic spaces and factored
spheres for which there would be more topological and analytical excitement
involving, as they do, multiple connectedness and conical singularities.

Regarding spectral spinors, our previous method involved fitting a
conjectured general form for the coefficients using specific numerical
values. The present technique establishes both this general form and
the actual expressions in one operation.
\section{\bf Acknowledgment}

We wish to thank Peter Gilkey for help regarding the algebra behind reference
[\putref{BBG}], and thanks also to Michael Bordag for interesting discussions.
KK was supported by the DFG under contract number Bo1112/4-1.

\section{\bf Appendix A.}
In this appendix we derive the form (\puteqn{finser}) for the coexact
sphere $\zeta$--function.
The most important property for the proof is the

\noindent{\bf Lemma.}
$$\eqalign{
\sum_{m=p+1}^d \left( m-1 \atop p \right) \zeta_{{\cal B}}( 2s; (a+1)/2 \mid {\bf 1}_m )=&
-\sum_{m=p}^{d-1} \left( m-1 \atop p-1 \right) \zeta_{{\cal B}}( 2s; (a-1)/2
\mid {\bf 1}_m ){}\cr
&\hspace{-2cm}+\left( d-1 \atop p \right) \zeta_{{\cal B}}( 2s; (a-1)/2\mid{\bf 1}_d ).\cr}
\eql{l2}$$
\noindent{\bf Proof.}

Rewrite the left-hand side of (\puteqn{l2}) using the difference relation,
[\putref{Barnesa}],
$$
\zeta_{{\cal B}}( s; (a+1)/2\mid{\bf 1}_m )-\zeta_{{\cal B}}( s; (a-1)/2\mid {\bf 1}_m ) = -\zeta_{{\cal B}}( s;
(a-1)/2\mid {\bf 1}_{m-1})
\eql{l1}
$$
and then combine terms using the binomial recursion (\puteqn{brecurs}).

Now we are set to show the following recursive property of the
$p$-form $\zeta$--function\ (\puteqn{spcoze})

\noindent {\bf Lemma.}
$$
\tilde \zeta_p^{S^d} (s)
+ \tilde \zeta_{p-1}^{S^{d-2}}(s)=\left( d-1 \atop p \right)\tilde
\zeta_0^{S^d}(s)+\delta_{p,1}\bigg({d-3\over2}\bigg)^{-2s}
+\delta_{p,0}\bigg({d-1\over2}\bigg)^{-2s}
 \eql{l3} $$
for $0\le p\le (d-1)/2$.

\noindent {\bf Proof.}

Starting from (\puteqn{spcoze}) and (\puteqn{zetzeroin}), use of (\puteqn{l2})
gives first
$$\eqalign{
\tilde\zeta_p^{S^d} (s) =&  -\sum_{m=p}^{d-1} \left( m-1 \atop p-1
\right) \zeta_{{\cal B}}( 2s; (d-1)/2\mid{\bf 1}_m ) +\left( d-1 \atop p \right) \zeta_{{\cal B}}( 2s;
(d-1)/2\mid{\bf 1}_d ){}\cr
&+\sum_{m=d-p}^d \left( m-1 \atop d-p-1\right) \zeta_{{\cal B}}( 2s;
(d+1)/2 \mid {\bf 1}_m )+\delta_{p,0}\bigg({d-1\over2}\bigg)^{-2s}.\cr}
$$
Separate the $m=d$ term from the last sum, then
$$\eqalign{
\tilde\zeta_p^{S^d} (s) =& \left( d-1 \atop p \right)
\bigg( \zeta_{{\cal B}}( 2s; (d+1)/2\mid {\bf 1}_d) +\zeta_{{\cal B}}( 2s; (d-1)/2\mid {\bf 1}_d \bigg)\cr
&-\sum_{m=p}^{d-1} \left( m-1 \atop p-1\right)
\zeta_{{\cal B}}( 2s; (d-1)/2\mid{\bf 1}_m )+\delta_{p,0}\bigg({d-1\over2}\bigg)^{-2s}\cr
&+\sum_{m=d-p}^{d-1}\left(m-1\atop d-p-1\right)\zeta_{{\cal B}}( 2s; (d+1)/2
\mid {\bf 1}_m ).\cr}
$$
Next, combine the $m = d-1$ terms of both sums using (\puteqn{l1}),
$$\eqalign{
\tilde\zeta_p^{S^d} (s) =&\left( d-1 \atop p \right)
\tilde\zeta_0^{S^d}(s) -\sum_{m=p}^{d-2} \left( m-1 \atop p-1\right)
\zeta_{{\cal B}}( 2s; (d-1)/2\mid{\bf 1}_m )\cr
 &+\sum_{m=d-p}^{d-2}\left( m-1 \atop d-p-1\right)
\zeta_{{\cal B}}( 2s; (d+1)/2 \mid {\bf 1}_m )\cr
&- \left( d-2 \atop p-1 \right) \zeta_{{\cal B}}( 2s; (d-1)/2\mid{\bf 1}_{d-2})
+\delta_{p,0}\bigg({d-1\over2}\bigg)^{-2s}.\cr}
$$
Apply (\puteqn{l2}) once more in the second sum and then the
binomial recursion, (\puteqn{brecurs}), to reach Lemma (\puteqn{l3}).
Finally, applying this Lemma $p+1$ times, for $p<(d-1)/2$,
produces the important relation ({\puteqn{finser}) in the main text. No doubt
this also follows from a direct analysis of the degeneracies.

A comment on these equations is perhaps necessary. Setting $p=0$ in
(\puteqn{l3}) we must recognise that $\tilde\zeta_{-1}^{S^{d-2}}(s)$ equals
$\big((d-1)/2\big)^{-2s}$ for consistency. This follows
easily from (\puteqn{spcoze}). In general, $\tilde\zeta_{-p}^{S^{d}}(s)=
\big((d-1+2p)/2\big)^{-2s}$.

\section{\bf Appendix B.}
In this appendix are listed some selected derivatives at zero of the
coexact zeta
function for absolute boundary conditions and with ${\cal M}$ the ball.

\noindent In $d=2$,
$$\eqalign{
{\zeta_0'}^{{\cal M}}=&-{{15}\over {32}} - {{\log2}\over {12}} + \log3
 - {{3\,\zeta'(-2)}\over 4} + {{5\,\zeta'(-1)}\over 2} + \zeta'(0)\cr
{\zeta_1'}^{{\cal M}}=&-{1\over {16}} + {{11\,\log2}\over 6}
- {{3\,\zeta'(-2)}\over 2} +
   3\,\zeta'(-1) - \zeta'(0)\cr
{\zeta_2'}^{{\cal M}}=&-{3\over {32}} - {{\log2}\over {12}}
- {{3\,\zeta'(-2)}\over 4} + {{\zeta'(-1)}\over 2}.\cr}
$$
In $d=3$,
$$\eqalign{
{\zeta_0'}^{{\cal M}}=&
  -{{1213}\over {4320}} +{{151\,\log2}\over {90}}
 + {{\zeta'(-3)}\over 3} + {{\zeta'(-2)}\over 2} +
   {{13\,\zeta'(-1)}\over 6} + \zeta'(0)\cr
{\zeta_1'}^{{\cal M}}=&{{5989}\over {10080}} - {{19\,\log2}\over {30}}
+ \zeta'(-3) +{{\zeta'(-2)}\over 2} -{{3\,\zeta'(-1)}\over 2}-\zeta'(0)\cr
{\zeta_2'}^{{\cal M}}=&
  -{{507}\over {1120}} + {{7\,\log2}\over {10}}
+\zeta'(-3)-{{\zeta'(-2)}\over 2}-{{7\,\zeta'(-1)}\over 2} +2\,\zeta'(0)
\cr
{\zeta_3'}^{{\cal M}}=&{{173}\over {30240}} + {{\log2}\over {90}}
+{{\zeta'(-3)}\over3}-{{\zeta'(-2)}\over 2}+{{\zeta'(-1)}\over 6}
. \cr  }
$$

\noindent In $d=4$,
$$\eqalign{
{\zeta_0'}^{\cal M} =&
  -{{25381}\over {46080}} + {{17\,\log2}\over {2880}} + \log5
 - {{5\,\zeta'(-4)}\over {64}} + {{23\,\zeta'(-3)}\over {48}}\cr
&\hspace{**************} +
   {{47\,\zeta'(-2)}\over {32}} + {{103\,\zeta'(-1)}\over {48}} +
   \zeta'(0)\cr
{\zeta_1'}^{\cal M} =&
  {{5803}\over {11520}} + {77\log2\over720}-\log3 -
   {{5\,\zeta'(-4)}\over {16}} + {{19\,\zeta'(-3)}\over {12}}\cr
&\hspace{**************} +{{17\,\zeta'(-2)}\over 8}-{{25\,\zeta'(-1)}
\over{12}} - \zeta'(0)\cr
{\zeta_2'}^{\cal M} =&
  {{209}\over {2560}} - {{863\,\log2}\over {480}} -
   {{15\,\zeta'(-4)}\over {32}} + {{15\,\zeta'(-3)}\over 8}\cr
&\hspace{**************} -
   {{3\,\zeta'(-2)}\over {16}} - {{21\,\zeta'(-1)}\over 8} +
   \zeta'(0)\cr
{\zeta_3'}^{\cal M} =&
  -{{2509}\over {11520}} +{77\log2\over720}- {{5\,\zeta'(-4)}\over {16}}
+{{11\,\zeta'(-3)}\over {12}}\cr
&\hspace{**************}
    - {{7\,\zeta'(-2)}\over 8} +
   {{19\,\zeta'(-1)}\over {12}} - \zeta'(0)\cr
{\zeta_4'}^{\cal M} =&
  {{47}\over {9216}} + {{17\,\log2}\over {2880}} -
   {{5\,\zeta'(-4)}\over {64}} + {{7\,\zeta'(-3)}\over {48}} -
   {{\zeta'(-2)}\over {32}} - {{\zeta'(-1)}\over {48}}.\cr}
$$
The $\zeta$--functions\ on the right are Riemann $\zeta$--functions.

As mentioned in the text, it is possible  to obtain the determinants in
any dimension $d$ and for any value of $p$ without difficulty.

\section{\bf Appendix C.}

In this appendix we outline a method for the calculation of integrals
of the type
$$\eqalign{
\int_a^b dw\, w^{n-1}\log\Gamma(w),\cr}
\eql{intneeded}
$$
needed in order to see explicitly the agreement of our results with those
of [\putref{ELV2}].

We start from the usual relation
$$
{\partial\over \partial w}\zeta(s,w)=-s\zeta(s+1,w),
\eql{100}$$
valid for any modified $\zeta$--function, such as $\zeta^{\cal N}}\def\nwilde{{\widetilde{\cal N}}_p(s)$ or the Barnes $\zeta$--function,
but here applied to the Hurwitz $\zeta$--function.

The differentiations are continued to give,
$$
{\partial^n\over \partial w^n}\zeta(s,w)=(-1)^n s(s+1)\ldots(s+n-1)\zeta(s+n,w)
=f_n(s)\zeta(s+n,w).
\eql{101}$$
Now differentiate with respect to $s$
$$
{\partial^n\over \partial w^n}\zeta'(s,w)=f_n'(s)\zeta(s+n,w)+f_n(s)\zeta'(s+n,w)
\eql{102}$$
and set $s=-n$
$$\eqalign{
{\partial^n\over \partial w^n}\zeta'(-n,w)&=f_n'(-n)\zeta(0,w)+f_n(-n)\zeta'(0,w)\cr
&=f_n'(-n)\zeta(0,w)+f_n(-n)(\log\Gamma(w)-\log\sqrt{2\pi}).\cr}
\eql{103}$$
Next, multiply by $w^{n-1}$ and integrate
$$\eqalign{
f_n(-n)\int_a^b dw\, w^{n-1}&\log\Gamma(w)=
\int_a^b dw\,w^{n-1}{\partial^n\over \partial w^n}\zeta'(-n,w)\cr
&+{f_n(-n)\over n}(b^n-a^n)\log\sqrt{2\pi}-f_n'(-n)\int_a^b
dw\, w^{n-1}\zeta(0,w).\cr}
\eql{104}$$
Define
$$
I_n(s)=\int_a^b dw\, w^{n-1}{\partial^n\over \partial w^n}\zeta'(s,w)
$$and integrate by parts to get the recursion
$$
I_n(s)=w^{n-1}{\partial^{n-1}\over \partial w^{n-1}}\zeta'(s,w)\bigg|_a^b-(n-1)I_{n-1}(s)
\eql{105}$$
which is trivially continued using $I_1(s)=\zeta'(s,b)-\zeta'(s,a)$

From (\puteqn{102}) we have
$$
w^{m}{\partial^{m}\over \partial w^{m}}\zeta'(-n,w)=
w^{m}\big(f_{m}'(-n)\zeta(-n+m,w)+f_{m}(-n)\zeta'(-n+m,w)\big)
\eql{106}$$
where
$$\eqalign{
f_m(-n)=m!\comb nm,\quad f'_m(-n)=m!\comb nm \sum_{k=1}^m{1\over m-n-k}
.\cr}
\eql{anja}$$
Using the recursion (\puteqn{105}) $n$ times, and also (\puteqn{anja}), the
following result is found for the integral (\puteqn{intneeded}),
$$\eqalign{
\int_a^b dw\, & w^{n-1}\log\Gamma(w)=
\bigg\{{1\over 2n} \bigg[
\log(2\pi) +\sum_{k=1}^n{1\over k}
\bigg] w^n -{1\over n+1} w^{n+1} \sum_{k=1}^n{1\over k}
\cr
&\sum_{l=1}^n {(-1)^{l-1} \over l} \comb{n-1}{l-1} w^{n-l}
\bigg[\zeta' (-l,w)-\zeta(-l,w)\sum_{k=1}^{n-l}
{1\over k+l} \bigg] \bigg\}\bigg|_a^b.\cr}
\eql{intfinal}
$$
The simplest case, when $n=1$, is given in [\putref{EandR}] and
an integral related to (\puteqn{intfinal}) for $n=2$ can be found in
Nash and O'Connor, [\putref{NandOC}] eqn.(C.30).

If one of the limits in (\puteqn{intneeded}) is zero, the last sum in
(\puteqn{intfinal}) has to be interpreted as the limit as $w\to 0$ and,
in this case, we use
$$\eqalign{
\zeta (s,w) =w^{-s} +\zeta (s,1+w)\cr}
\eql{zetarec}
$$ to see that
$$
\lim_{w \to 0} w^{n-l}\bigg[ \zeta' (-l,w) -\zeta (-l,w)
\sum_{k=1}^{n-l} {1 \over k+l} \bigg] =0
$$
for $l<n$, whereas
$$
\lim_{w\to 0} \zeta' (-n,w) = \zeta ' (-n)
$$
for $n\geq 1$.

Let us now give some examples of the general formula (\puteqn{intfinal}).
To compare with $\zeta_0'(0)$ in $d=3$ of [\putref{ELV2}], we need
$$
\int_0^1 dw\,(w-1)\log\Gamma(w),
$$
which is obtained by setting $n=1$ and $n=2$.
For $n=1$, we get the well known formula ([\putref{Erdelyi}], 1.9.1 (18))
$$
\int_0^1dw\,\log\Gamma(w)={1\over2}\log(2\pi)=-\zeta'(0)
\eql{107}$$
which is the essential part of Raabe's formula.

For $n=2$
$$
\int_0^1 dw\,w\log\Gamma(w)=-{1\over12}-{1\over2}\zeta'(0)+\zeta'(-1).
\eql{108}$$
The integral term in [\putref{ELV2}] is therefore
$$
2\int_0^1 dw\,(w-1)\log\Gamma(w)= -{1\over6}+2\zeta'(-1)+\zeta'(0).
$$
In other dimensions, generally $a$ and $b$ take integer or half-integer
values. Using (\puteqn{zetarec}) the integrals can always be expressed
in terms of derivatives of the Riemann $\zeta$--function. We meet some
known integrals such as([\putref{Voros,EandR}])
$$
\int_1^{3/2}dw\, \log\Gamma(w)=-{3\over8}-{13\over24}\log2-{3\over2}\zeta'(-1)-
{1\over2}\zeta'(0)
\eql{109}$$
and some lesser known ones, {\it e.g. }
$$
\int_0^{3/2}dw\,w\log\Gamma(w)=-{1\over2}-{11\over 16}\log2+{7\over 8}\zeta'(-2)
-{3\over4}\zeta'(-1)-{9\over8}\zeta'(0)
\eql{110}$$
generalisations of which can be found using (\puteqn{intfinal}).

Finally we point out that a generalisation of Raabe's formula to the
multiple $\Gamma$-function is derived by Barnes [\putref{Barnesa}] using the
same ingredients as in the above.

\section{\bf Appendix D.}
The identity (\puteqn{siden1}) will now be proved.
In terms of the spinor base $\zeta$--function\ it reads
$$
\Res\,\,\zn 1/2\right) -\Res\,\,\zn 1\right) -2\sum_{i=1}^{d-2} \Res\,\,\zn 1+i/2\right)
D_i' (1) =0   .
$$
Using knowledge of the residues in terms of the Bernoulli functions,
[\putref{Barnesa}], one has
$$\eqalign{
0=\frac{B_{d-1}^{(d)} \left((d-1)/ 2\right)}{(d-1)!}
&+\frac{B_{d-2}^{(d)} \left((d-1)/ 2 \right)}{(d-2)!}\cr
& +2\sum_{i=1}^{d-2} (-1)^i D_i' (1)\, \frac{B_{d-2-i}^{(d)}
\left((d-1)/2\right)}{(1+i)! (d-2-i)!},\cr}
\eql{siden2}$$
which is a standard recursion formula for the generalized Bernoulli
functions as will now be shown.

We first note that $D_i'(1)=(-1)^i/2$. This value is obtained by comparing
the
small $z$ approximation of the Bessel function $I_\nu(\nu z)$ with Olver's
asymptotic form. Thus, working always to order $z^2$,
$$
\ln I_\nu(\nu z)\sim \nu\ln\big(\nu z/2\big)-\ln\Gamma(1+\nu)+\ln\bigg(
1+{1\over4(\nu+1)}(\nu z)^2\bigg).
$$
Expanding the log,
$$
\ln\bigg(1+{1\over4(\nu+1)}(\nu z)^2\bigg)\sim{1\over4} z^2{\nu\over1+1/\nu}
={1\over4}z^2\nu\sum_{n=0}^\infty(-1)^n \nu^{-n}
$$
and also
$$
\ln\Gamma(1+\nu)\sim \nu\ln \nu-\nu+{1\over2}\ln2\pi\nu-\sum_{k=1}^\infty
{\zeta_R(-k)\over k}\nu^{-k}.
$$
The Bessel asymptotics are
$$\eqalign{
\ln I_\nu(\nu z)&\!\sim\! -{1\over2}\ln2\pi\nu\!-\!{1\over4}\ln(1+z^2)\!
+\!\nu
\bigg(\!\sqrt{1+z^2}\!+\!\ln\big({z\over1+\sqrt{1+z^2}}\big)\!\bigg)\!+\!
\sum_{n=1}^\infty\!{D_n(t)\over\nu^n}\cr
&\sim-{1\over2}\ln2\pi\nu-{1\over4}z^2+\nu\big(1+{1\over4}z^2+\ln(z/2)\big)
+\sum_{n=1}^\infty{D_n(1)-z^2D_n'(1)/2\over\nu^n}.\cr}
$$

Comparing, we obtain the quoted value of $D_n'(1)$, and also that of
$D_n(1)$
met in our previous work. The higher derivatives can clearly be found by
carrying the expansion further.

Substituting these values of $D_n'(1)$ into (\puteqn{siden2}) yields the
condition
$$
\sum_{j=0}^{d-1}\comb{d-1}j B^{(d)}_{d-1-j}\big((d-1)/2)\big)=0
\eql{cond}
$$
which is a special case of the recursion relation ([\putref{norlund2}]
eqn.(11) p161)
$$
\sum_{j=0}^\nu\comb\nu j\omega^jB^{(n)}_{\nu-j}\big(x\mid\bom\big)=
\omega\nu B^{(n-1)}_{\nu-1}\big(x\mid\bom'\big),\quad \bom=(\bom',\omega)
\eql{nrec}$$
involving the general Bernoulli function. Setting $\bom={\bf 1}$,
$\nu=d-1$ and $n=d$, this becomes
$$\eqalign{
\sum_{j=0}^{d-1}\comb{d-1}jB^{(d)}_{d-1-j}(x)&=(d-1)B^{(d-1)}_{d-2}(x)\cr
&=(d-1)(x-1)(x-2)\ldots(x-d+2).\cr}
\eql{srec}$$

If $d$ is odd, $(d-1)/2$ is integral and, since $1<(d-1)/2\le d-2$ for
$d>1$, the right-hand side of (\puteqn{srec}) vanishes for all odd $d$
when $x=(d-1)/2$, as required to show (\puteqn{cond}).

For even $d$, $=2q$, (\puteqn{srec}) allows one to give the general form
of the coefficient
$$
A^{(S)}_{(D-2)/2}(D)=(-1)^q\big(q-{1\over2}\big)\bigg(\big(q-{3\over2}\big)
\ldots {3\over2}\bigg)^2,\quad D=2q+1.
\eql{gc}$$

A similar result should hold for the monopole. Put $\omega=a$ in
(\puteqn{nrec}).

\section{\bf References}
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\end{putreferences}

\bye

redpajama_set_name: RedPajamaArXiv


text:
require 'rails_helper'

describe Multilang::Searcher do

  describe '#translation_keys' do
    before do
      create(:language, is_default: true)
      @translation_key = create(:translation_key)
      @translation_key.translations.first.update_attribute(:value, 'Hello, World')
      create(:translation_key, key: 'world')
    end

    it 'will find by key' do
      searcher = build_searcher(key: @translation_key.key)
      expect(searcher.translation_keys.count).to eq(1)
    end

    it 'will find by translation value' do
      searcher = build_searcher(q: 'hello')
      expect(searcher.translation_keys.count).to eq(1)
    end

    it 'will not find by completed translations' do
      searcher = build_searcher(complete: true)
      expect(searcher.translation_keys.count).to eq(0)
    end

    it 'will find by completed translations' do
      searcher = build_searcher(complete: false)
      expect(searcher.translation_keys.count).to eq(2)
    end
  end

  def build_searcher(attrs = {})
    Multilang::Searcher.new(Multilang::Search.new(attrs))
  end
end

redpajama_set_name: RedPajamaGithub