Funny pictures of sexps

[jaydfox]

To my knowledge, one cannot construct a Fourier series that is smooth in both imaginary directions. Only one direction can be smooth, and only if one allows complex results for real inputs. Otherwise, neither direction is smooth.

If you know otherwise, then my argument would seem to fall apart.

Actually, I take that back. One could construct an infinite sequence of polar singularities at the integers. For example, \( f(x) = \sum_{n=-\infty}^{+\infty} \left(\frac{1}{(z-n)^2}\right) \). As we move infinitely far from the real axis, the effects of these singularities diminishes rapidly enough that, despite the cumulative effects of an infinite number of them, we should get 0 as we move infinitely far from the real axis. And being cyclic with a period of 1 on the real axis, this function would probably admit a Fourier series expansion.

However, this method introduces singularities on the real axis, so I stand by my original assertion (which included a proscription against introducing new singularities in the principal branch). You can't create a non-constant function which goes to a constant value at infinity without creating a singularity somewhere... Can you? (Not being sarcastic, this is an honest question. As far as I can remember, this is true, but I welcome being shown that I'm mistaken on this point.)