In complex integration my book uses that some limits go to zero as R goes to infinity. However I do not now how to show this, these two limits are:
$e^{-\pi(R^2+2iRy-y^2)}$, where y is a real number that can vary from 0 to a real numer b.
The second limit is $e^{-R^2e^{2it}}R$. Normally I would have used l'hopitals rule, but when I have complex number, it isn't clear how to do this, t is a number from 0 to $\pi/4$.
For the firs limit I can try something like this:
$|e^{-\pi(R^2+2iRy-y^2)}|\le K|e^{-\pi(R^2+2iRy)}|=K|e^{-\pi R(R+2iy)}|$, where K is a real number but how do I show that this goes to 0?, the problem is that I have a complex number in the exponent.
For the second one I get what I need if I can show that $|e^{-R^2e^{2it}}|\le L|e^{-R^2}|$, because then I can work with real numbers, but how do I do this?