Limit question $\lim_{k\to 0}\int_0^{\pi}e^{ik e^{i\theta}} d\theta$ How would you formally show that the following limit converges to $\pi$?
$$\lim_{k\to 0^+}\int_0^{\pi}e^{ik e^{i\theta}} d\theta$$.
Any help much appreciated :)

I do not know if this is the kind of answer you expect.
What I should do is to develop the integrand as a Taylor series around $k=0$. This gives $$e^{i k e^{i \theta }}=1+i e^{i \theta } k-\frac{1}{2} e^{2 i \theta } k^2+O\left(k^3\right)$$ Now, integration $$\int e^{i k e^{i \theta }}d\theta=\theta +e^{i \theta } k+\frac{1}{4} i e^{2 i \theta } k^2+O\left(k^3\right)$$  Using bounds and the famous $e^{i\pi}+1=0$$$\int_0^\pi e^{i k e^{i \theta }}d\theta=\pi -2 k+O\left(k^3\right)$$ which shows the limit and how it is approached.
