Integrating $\int_{0}^{x} e^{ie^{\frac{s}{2}}}ds$

Question

How would one proceed to evaluate $\int_{0}^{x} e^{ie^{\frac{s}{2}}}ds$? I was considering a change of variables $p=\sqrt{-i}e^{\frac{s}{4}}$, so that I could express the function in terms of $\mathscr{Erf}$, though the expansion for $ds$ would not be tidy.

Answer 1

Hint:

$$ e^{ie^{t/2}}=\sum^\infty_{n=0}(i)^ne^{nt/2}/n! $$

Exchange order of integration and summation (this need some justification, but this is valid here) $$ \int^x_0e^{ie^{t/2}}\,dt=\int^x_0\Big(\sum^\infty_{n=0}\frac{i^n}{n!}e^{nt/2}\Big)\,dt=\sum^\infty_{n=0}\frac{i^n}{n!}\int^x_0e^{nt/2}\,dt $$

Can you finish from here?