- #1
OscarAlexCunning
- 2
- 1
Say we have ##P_k(z)## a family of entire functions, and they depend analytically on ##k## in ##\Delta##. Assume ##P_k(z)## is nonzero on ##S^1## for all ##k##. How do I see that for each ##t \ge 0##, we have that$$\sum_{|z| < 1, P_k(z) = 0} z^t$$is an analytic function of ##k##? Here, the zeros of ##P_k(z)## are regarded with multiplicity.
Now, I can show this myself, but I am wondering if you guys have any alternative solutions.
My way of doing this. Via the residue theorem, for each ##k## in ##\Delta## we have$$\sum_{|z| < 1, P_k(z) = 0} z^t = {1\over{2\pi i}} \int_{\partial \Delta} {{P_k'(z)z^t}\over{P_k(z)}}dz.$$This integral representation makes it evident that our function of interest is analytic in ##k## by, for instance, the theorem of Morera.
Now, I can show this myself, but I am wondering if you guys have any alternative solutions.
My way of doing this. Via the residue theorem, for each ##k## in ##\Delta## we have$$\sum_{|z| < 1, P_k(z) = 0} z^t = {1\over{2\pi i}} \int_{\partial \Delta} {{P_k'(z)z^t}\over{P_k(z)}}dz.$$This integral representation makes it evident that our function of interest is analytic in ##k## by, for instance, the theorem of Morera.