Good afternoon.
I am still studying on Bessel functions, and am currently struggling with the proof of Graf's addition theorem, to be found in Watson 11.3.
My main concern is the middle section, where he performs the transformation
$$(Z - z e^{-i\phi}) t = \varpi u \qquad (Z - z e^{i\phi}) / t = \varpi / u$$ with $$\varpi = \sqrt{Z^2 + z^2 -2Zz \cos(\phi)}.$$
He then argues that that branch of the root is taken which makes $\varpi \to Z$ as $z \to 0$. Then, he claims that the phase of $\varpi/Z$ is always an acute angle (positive or negative). Notice that $|z e^{\pm i \phi}| \le |Z|$ is enforced. This is the first point that I don't quite understand, why the phase of this expression is always acute.
Then, secondly, I don't quite understand how the integration contour is transformed. At first, we have a contour coming from (and ending at) $-\infty \exp(-i \alpha)$, where $\alpha$ is the argument of $Z$, and looping around the origin once. Then, after the transformation, the contour starts and ends at $-\infty \exp(-i \beta)$, $\beta$ being the argument of $\varpi$, and again loops about the origin. Why is that?
Any clarification is greatly appreciated.