7
The problem is probably that the formula with the Bessel function only applies to some m and in general it will be more complicated. Take a look at the following graph.
Show[Plot[NIntegrate[Cos[m* k] Exp[Cos[k]], {k, 0, 2 Pi}], {m, 0, 5}, PlotStyle -> Red], Plot[2*Pi*BesselI[m, 1], {m, 0, 5}]]
2
As n is integer you can try
Table[Integrate[(n (1 + n) (1 + 2 n) (3 + 2 n)
(-HypergeometricPFQ[{1, 1, 1 - 2 n, 4 + 2 n}, {2, 3, 3}, 1] +
t HypergeometricPFQ[{1, 1, 1 - 2 n, 4 + 2 n}, {2, 3, 3}, t])
Log[t])/(-1 + t), {t, 0, 1}], {n, 1, 10}]
for n up to 10 or higher.
The result -10Pi^2/3 for n=1 is just the integral over the term with the first ...
2
This may give you what you want.
expr = (2^(-3 + 4/m) E^(-((3 I \[Pi])/m)) (-1 +
E^((2 I \[Pi])/m))^3 Gamma[(-2 + m)/m] Gamma[(-1 + m)/
m] Gamma[
1/m + (I w3)/
2] (Beta[(-1 + m)/m, 1/(2 m) + 1/4 I (w1 + w3)] Gamma[
1/4 (4 - 2/m - 2 I w1 + I w3)] Gamma[(2 + 2 I m w1 -
I m w3)/(4 m)] -
Beta[(-1 + m)/m,...
1
Maybe the following? I didn't understand all the extra variables, so I stripped down the code more. In a complicated case, you might need Dynamic@Refresh instead of just Dynamic; see What is the point of Refresh if Dynamic has an UpdateInterval option? and related Q&A on Refresh. You'd probably have to put the output in a docked cell, since the FE ...
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