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
Famous quotes containing the words solution, heat and/or equation:
“Coming out, all the way out, is offered more and more as the political solution to our oppression. The argument goes that, if people could see just how many of us there are, some in very important places, the negative stereotype would vanish overnight. ...It is far more realistic to suppose that, if the tenth of the population that is gay became visible tomorrow, the panic of the majority of people would inspire repressive legislation of a sort that would shock even the pessimists among us.”
—Jane Rule (b. 1931)
“And suddenly, to be dying
Is not a little or mean or cheap thing,
Only wearying, the heat unbearable ...”
—John Ashbery (b. 1927)
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)