Jacobi Theta Function
While exploring about the Riemann Zeta function I stumbled across another interesting function known as Jacobi Theta function. It is named after the German mathematician Carl Gustav Jacob Jacobi( an interesting name indedeed )
The Jacobi theta function is defined as :
This function has many applications in the complex analysis and also comes into picture while delaing with Riemann zeta function. So in this article I want to explore this function and derive an imporatant result at the end.
P.S. (I have tried a different editor for writing the equations this time:))
So lets dive in!
Poisson Summation
Now before getting to the actual derivation we need a result known as The Poisson Summation formula. This formula states that if there is a Schwartz function f which is periodic then the following is true:
Lets derive this by taking a function f defined as
This shows that it is a 1-periodic function. Lets see what happens when we find its Fourier transform,
Lets use simple substitution,
And we will get,
As we know that
Dealing with the Integral
Here in this case,
Substituting the function we get,
Now completing the square,
Now lets just focus on the integral only. Lets make the following substitution
After making the above substitution, the limits of the integral also changes and they become the follows:
So finally the integral becomes something like this
Now this integral in the brackets looks quite weird with those lower and upper limits not making any sense. So lets deal with it firsthand
In order to make sense of this we need to use Complex analysis. So lets begin by writing the integrand as complex function h(z) and rewriting the integral as follows:
You may be wondering why we replaced ∞ with R ?
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Well the reason is simple, just to avoid writing infinity. Remember how we can write improper integrals in the form of proper integrals using limits. Lets say we have an integral like
Then we can also write it as follows
We just replace the infinite limits with a variable and then take the limit tends to infinity. This makes it easier to deal with integrals of this form as we just compute everything inside the brackets as a usual definite integral and then find the limits.
The same thing is being done here, we are just using R now but later on we will take its limit. Also using a finite variable makes it easy to define a contour for complex integration. As we know while performing integration on complex functions we need to define a particular path or curve as there can be infinitely many curves joining the end points in the complex plane. So in this case the curve is a simple rectangle and by convention we are performing the operation in counter-clockwise manner.
But from the residual theorem we get,
As the function ℎ(z) has no poles in the analytic region thus the number of residues is also zero.
So we can rewrite it as,
Lets consider the first integral
Lets substitute
The limits become
Similarly we can do for the third integral,
Substituting,
The limits become
So this leaves us with the following
Now we can replace back R with ∞ and we will get,
Now substituting in the original equation,
Here we can see that the integral inside the brackets is a Gaussian integral and we know its value, so we can substitute that value:
Thus finally we can write
But wait! we can write the RHS in terms of the theta function as follows:
Thus we finally arrive at the functional equation form of the Jacobi theta function.
