ProofOfJacobisIdentityForVarthetaFunctions 2004-10-29 23:46:38 2005-02-19 14:20:19 Proof Jacobi's identity for $\vartheta$ functions 0 % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here We start with the Fourier transform of $f(x) = e^{i \pi \tau x^2 + 2 i x z}$: $$\int_{-\infty}^{+\infty} e^{i \pi \tau x^2 + 2 i x z} e^{2 \pi ixy} \, dx = (- i \tau)^{-1/2} e^{-i {(z + \pi y)^2 \over \pi \tau}}$$ Applying the Poisson summation formula, we obtain the following: $$\sum_{n=-\infty}^{+\infty} e^{i \pi \tau n^2 + 2 i n z} = (- i \tau)^{-1/2} \sum_{n=-\infty}^{+\infty} e^{-i {(z + \pi n)^2 \over \pi \tau}}$$ The left hand \PMlinkescapetext{side} equals $\vartheta_3 (z \mid \tau)$. The right hand \PMlinkescapetext{side} can be rewritten as follows: $$\sum_{n=-\infty}^{+\infty} e^{-i {(z + \pi n)^2 \over \pi \tau}} = e^{-i {z^2 \over \pi \tau}} \sum_{n=-\infty}^{+\infty} e^{-i {\pi n^2 \over \tau} - {2 i n z \over \tau}} = e^{-i {z^2 \over \pi \tau}} \vartheta_3 (z / \tau \mid -1 / \tau)$$ Combining the two expressions yields $$\vartheta_3 (z \mid \tau) = e^{-i {z^2 \over \pi \tau}} \vartheta_3 (z / \tau \mid -1 / \tau)$$