Mathematica will not compute this integral

Question

I am trying to compute the following integral, but regardless of FullSimplify or Assumptions, Mathematica seems unable to compute it. I have already tried computing the indefinite version and evaluate that at the end points, but mathematica is unable to compute that one also. Any help would be greatly appreciated.

f[t_] := Exp[I c (t/2 + Sin[2 \[Omega] t]/(4 \[Omega]))]

tmp = Integrate[f[t], {t, 0, (2 \[Pi])/\[Omega]}] // FullSimplify

Answer 1

One easy way is: $$\int_0^{\frac{2 \pi }{\omega }} e^{\frac{i c t}{2}+\frac{i c \sin (2 t \omega )}{4 \omega }} \, dt=\int_0^{\frac{2 \pi }{\omega }} e^{\frac{i c t}{2}} \sum _{n=0}^{\infty } \frac{\left(\frac{i c \sin (2 t \omega )}{4 \omega }\right)^n}{n!} \, dt=\sum _{n=0}^{\infty } \int_0^{\frac{2 \pi }{\omega }} \frac{e^{\frac{i c t}{2}} \left(\frac{i c \sin (2 t \omega )}{4 \omega }\right)^n}{n!} \, dt$$ Then:

$Version
(*"14.1.0 for Microsoft Windows (64-bit) (July 16, 2024)"*)

Sum[Integrate[E^((I c t)/2)*((I c Sin[2 t \[Omega]])/(4 \[Omega]))^n/n!, {t, 0, (
2*\[Pi])/\[Omega]}, 
Assumptions -> {c > 0, \[Omega] > 0, n >= 0}], {n, 0, 
Infinity}] // FullSimplify

(*Closed-Form see below*)

$$\color{blue}{\int_0^{\frac{2 \pi }{\omega }} e^{\frac{i c t}{2}+\frac{i c \sin (2 t \omega )}{4 \omega }} \, dt=\\\frac{2 i \, _1F_2\left(1;1-\frac{c}{8 \omega },1+\frac{c}{8 \omega };-\frac{c^2}{64 \omega ^2}\right)}{c}-\frac{2 i e^{\frac{i c \pi }{\omega }} \, _1F_2\left(1;1-\frac{c}{8 \omega },1+\frac{c}{8 \omega };-\frac{c^2}{64 \omega ^2}\right)}{c}-\frac{2 i c \, _1F_2\left(1;\frac{3}{2}-\frac{c}{8 \omega },\frac{3}{2}+\frac{c}{8 \omega };-\frac{c^2}{64 \omega ^2}\right)}{c^2-16 \omega ^2}+\frac{2 i c e^{\frac{i c \pi }{\omega }} \, _1F_2\left(1;\frac{3}{2}-\frac{c}{8 \omega },\frac{3}{2}+\frac{c}{8 \omega };-\frac{c^2}{64 \omega ^2}\right)}{c^2-16 \omega ^2}}$$

Check:

f[c_, \[Omega]_] := NIntegrate[
E^((I c t)/2 + (I c Sin[2 t \[Omega]])/(4 \[Omega])), {t, 0, (
2*\[Pi])/\[Omega]}];

g[c_, \[Omega]_] := -((2 I (-1 + E^((
  I c \[Pi])/\[Omega])) (HypergeometricPFQ[{1}, {1 - c/(
     8 \[Omega]), 1 + c/(8 \[Omega])}, -(c^2/(64 \[Omega]^2))] - (
  c^2 HypergeometricPFQ[{1}, {3/2 - c/(8 \[Omega]), 
     3/2 + c/(8 \[Omega])}, -(c^2/(64 \[Omega]^2))])/(
  c^2 - 16 \[Omega]^2)))/c);

{f[2, 2], g[2, 2] // N} // Chop
(*{0. + 2.10078 I, 0. + 2.10078 I}*)