- #1
Grothard
- 29
- 0
Homework Statement
Let [itex] f(z) = \sum_{n =-\infty}^{\infty} e^{2 \pi i n z} e^{- \pi n^2}.[/itex] Show that [itex] f(z+i) = e^{\pi} e^{-2\pi i z}f(z). [/itex]
Homework Equations
Nothing specific I can think of; general complex analysis/summation techniques.
The Attempt at a Solution
[itex] f(z+i) = \sum_{n =-\infty}^{\infty} e^{-2 \pi n z} e^{2 \pi i n z} e^{- \pi n^2};[/itex] I can't factor the new term out of the sum because it contains an n. I feel like I might be missing some sort of summation identity that can accomplish this. I also tried completing the square, but it didn't really get me far.