Very curious question

[mike3]

Let's make another function that equals its own derivative. I'm very curious as to why this is happening!

\( g(s) = \sum_{n=-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(y+n)^2} \frac{s^y}{\Gamma(y+1)}dy \)


Differentiate and watch for your self!

Does this mean the function cannot converge? I know the integral converges, not sure about the summation though.

Using the other method I can easily create a function that converges for some domain... What's going on?

The summation does not look like it converges. Try graphing the integrand for s = 1 and look what happens as n increases.

Also, using a numerical integration from \( -8-n \) to \( 8-n \) (roughly centers around the "peak", at least for relatively small n), one can approximate the integral and see the divergence:

n = 1, s = 1: 0.38446
n = 2, s = 1: 0.042752
n = 3, s = 1: -0.082158
n = 4, s = 1: 0.26084
n = 5, s = 1: -0.83652
n = 6, s = 1: 2.2210
n = 7, s = 1: 2.4999
n = 8, s = 1: -149.51

So the sum of these values approaches no limit. While the values do shrink for negative \( n \), the sum also includes the problematic positive values.

Note that this numerical test is not a proof of divergence, but it strongly indicates that is what is happening.