## 5408 Reputation

17 years, 226 days
Munich, Germany

## Re: References...

@Christian Wolinski Unfortunately I do not have a specific reference (except classical books like Kraft or Dolgachev which have in mind the general theory). Perhaps it is better to look for lectures & exercises. Your group is the one generated by the permutation 1 -> 2, 2 -> 3, 3 -> 1 (which is of order 3 and cyclic - but you take only that permutation). May be you ask in a math forum. If you would have q=0 then there are known relations (the Newton identities)

## invariant theory?...

Just a thought: with some luck you may find answers within (classical) invariant theory (for permutations?)

## just a variation of that...

Let R be Kitonum's rational function. Then series shows the singularity using

MultiSeries:-series(R, y=1, 6) or
MultiSeries:-multiseries(R, y=1, 'exact_order')

It can be seen that S:= - ( (3/4/(1-y)^2) - 21/4/(1-y) ) is the singularity in y=1.

limit(R-S, y=1) gives 87/16 and confirms it (or use series again)

## Euler's formula...

https://en.wikipedia.org/wiki/Euler%27s_formula

## x=0...

For x=0 the term is not defined

## ahem...

@rcorless certainly you meant @vv , not me ...   :-)

## filename ?...

What is the filename? Have you tried a name having only ASCII letters and numbers? Have you contacted the technical support by mail?

NB: I am using Win, not Mac

## hm (2)...

'u_tilde*exp(I*omega*t)/2/Pi';
'eval(%, omega=-omega) + %';

simplify(%) assuming 0<= omega:
evalc(Re(%))  assuming 0<= omega:
#map(identify, %):
vRe:=unapply(%, omega, t);

Now one wants to integrate vRe from 0 to 40 (instead of infinity), giving a function it t.

vRe(omega, 1); plot(%, omega=14.5 .. 14.6); # (t=1 as an example)

That plotting shows something which probably needs a Cauchy Principal Value and it is no clear (for me) why summation is good enough (omega = 10 is ok, it cancels out)

## hm ......

I do not quite understand why one can cut off at 0 and 40 and why that summing is a reasonable or rough approximation.

The integrand has singularities - the denominator only depends on omega and has zeroes, omega = 10 gives a pole for example.

## pls provide as code...

You should post it as code and care for correct brackets

## suggestion...

I do not quite understand your sheet, but having Sum(f(n), n= a .. b) and setting g:= k -> f(5*k) then Sum(g(n), n= a .. b) might be what you are looking for. So write your summands as function f.

## likewise...

Likewise and not caring for the discontinuity:

G:=w -> Int(eval(A1+A2,omega=w),theta=0..2*Pi, method = _d01ajc, epsilon=1e-3);

Digits:=15;
plot(G, 0 .. 50);

## not my point...

My point is: Maple uses the conventions (or I get you wrong), I do not want to discuss conventions

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