A Solution To Heat Equation
The Jacobi theta function is the unique solution to the one-dimensional heat equation with periodic boundary conditions at time zero. This is most easily seen by taking z = x to be real, and taking τ = it with t real and positive. Then we can write
which solves the heat equation
That this solution is unique can be seen by noting that at t = 0, the theta function becomes the Dirac comb:
where δ is the Dirac delta function. Thus, general solutions can be specified by convolving the (periodic) boundary condition at t = 0 with the theta function.
Read more about this topic: Theta Function
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