Integral of $\exp⁡(-j\sin⁡(t))$ over $[0,T]$

Question

I have to solve the following integral : $$ \int_0^T \exp\left(-j \sin(t)\right) \,\mathrm{d}t $$ where $j=\sqrt{−1}$

How do I proceed to solve this?

Answer 1

$e^{i\sin t}$ is a $2\pi$-periodic function (and an entire function) with the following Fourier series:

$$ e^{i\sin t} = J_0(1) + 2i\sum_{n\geq 0}J_{2n+1}(1)\sin((2n+1)t)+2\sum_{n\geq 1}J_{2n}(1) \cos(2nt) \tag{1}$$ hence: $$ \int_{0}^{T}e^{i\sin t}\,dt = T\cdot J_0(1)+2i\sum_{n\geq 0}\frac{J_{2n+1}(1)}{2n+1}-2i\sum_{n\geq 0}\frac{J_{2n+1}}{2n+1}\cos((2n+1)T)+\sum_{n\geq 1}\frac{J_{2n}(1)}{n}\sin(2nT)\tag{2} $$ where $J_\nu$ is a Bessel function of the first kind, so that $J_\nu(1)$ behaves like $\frac{1}{\nu! 2^\nu}$ and $$ J_0(1)=\sum_{n\geq 0}\frac{(-1)^n}{4^n n!^2}\approx\frac{88}{115}.\tag{3}$$