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JACOBI’S BASIC THETA FUNCTION

Define   X 2 z = x + iy ∈ C θ(z, τ) = e2πinz+πin τ , where . τ = σ + it ∈ H n∈Z Our main use of the theta function will be that its restriction to z = 0 is a in τ, and the Euler– is essentially the of the modular form: this is one way that Riemann established the analytic contin- uation and functional equation of zeta. But this writeup will briefly describe how the theta function solves the . Theta functions in general are power- fully versatile. ’s Tata Lectures on Theta volumes are an excellent source of reading on this topic.

The magnitude of the θ(x + iy, σ + it) summand is

2 2 |e2πin(x+iy)+πin (σ+it)| = e−2πny−πn t = o(e−π|n|t) as |n| → ∞. Thus the series converges rapidly for any t > 0, thanks to the summand-component 2 eπin τ . Clearly θ(z, τ) is Z-periodic in z. (Its periodicity in τ, both stronger and more subtle than in z, will be discussed in a separate writeup.) Restrict the theta function to z = x ∈ R and τ = it with t > 0, and give the restriction θ(x, it)—which takes real values—its own name,

X 2πinx−πn2t uθ : R × R>0 −→ R, uθ(x, t) = e . n∈Z The idea is that x is a position variable and t a time variable, and the restriction visibly satisfies the heat equation, 2 ∂ uθ X 2 (x, t) = (−4π2n2)e2πinx−πn t ∂x2 n∈Z X 2 ∂uθ = 4π (−πn2)e2πinx−πn t = 4π (x, t). ∂t n∈Z Furthermore, the limiting initial time-value of u(x, t) is the Z-periodicized in the position variable, ( ∞ if x ∈ ) X 2πinx Z per lim uθ(x, t) = e = = δ (x). t→0+ 0 if x∈ / Z n∈Z Thus we have a periodic solution of the heat equation whose limiting initial condi- tion is particularly handy. The general heat equation is now easy to solve. Consider an arbitrary periodic initial time-value condition uo. Let vo be its restriction to [0, 1) extended by zero 1 2 JACOBI’S BASIC THETA FUNCTION to R, so that uo is its Z-periodicization, X uo(x) = vo(x + n). n∈Z Consider the convolution of uθ and vo, Z u = uθ ∗ vo : R × R>0 −→ R, u(x, t) = uθ(x − x,e t)vo(xe) dx.e R Differentiation under the integral sign shows that u again satisfies the heat equation, ∂2u Z ∂2u (x, t) = θ (x − x, t)v (x) dx ∂x2 ∂x2 e o e e R Z ∂u ∂u = 4π θ (x − x, t)v (x) dx = 4π (x, t). ∂t e o e e ∂t R Furthermore, the limiting initial-time value of u is Z lim u(x, t) = lim uθ(x − x, t)vo(x) dx t→0+ t→0+ e e e ZR per X = δ (x − xe)vo(xe) dxe = vo(x + n) = uo(x). R n∈Z Thus, modulo many informalities, the theta function solves the periodic heat equa- tion.