Using the Theta Function to Solve for a Jacobi-Related Equation

[Wizlem]

I was reading a book on the zeta function and came across this attributed to Jacobi. I have no idea where to find a source about this so maybe someone can give me some direction. Let

$$\psi(x) = {\sum}^{\infty}}_{n=1}e^{-n^2 \pi x}$$.

How do you show that

$$\frac{1+2\psi(x)}{1+2\psi(1/x)} = \frac{1}{\sqrt{x}}$$

 

[jackmell]

Try framing it in terms of the theta function:

$$\theta(t)=\sum_{n=-\infty}^{\infty}e^{-\pi n^2 t}$$

which has the functional equation:

$$\theta(t)=t^{-1/2} \theta(1/t)$$

Now, express your $\psi$ function in terms of this functional equation.

Also, see "Complex Analysis" by Stein and Shakarchi.