derivation of integral representations of Jacobi ϑ functions

By rearranging the Fourier series of cos(ux), one obtains the series


This equation which is valid for all real values of x such that -πxπ and all non-integral complex values of u. By comparison with the convergent series n=01/n2, it follows that this series is absolutely convergent. Note that this series may be viewed as a Mittag-Leffler partial fraction expansion.

Let y be a positive real number. Multiply both by 2ue-yu2 and integrate.


Because of the exponential, the integrand decays rapidly as ui± provided that u>0, and hence the integral converges absolutely. Make a change of variables v=u2


The contour of integration P is a parabolaPlanetmathPlanetmath in the complex v-plane, symmetric about the real axis with vertex at v=-1, which encloses the real axis. Its equation is v+1=2(v)2

Let Sm (m is an integer) be the straight line segment joining the points v=(i+m+1/2)2 and v=(i-m-1/2)2. Along this line segmentMathworldPlanetmath, we may bound the integrand in absolute valueMathworldPlanetmathPlanetmathPlanetmath as follows:


where vm=m2+m-3/4 is the point of intersectionMathworldPlanetmath of Sm with the real axis. To proceed further, we break up the last summation into two parts.

Since the squares closest in absolute value to vm are m2 and (m+1)2=m2+2m+1, it follows that |vm-n2||m-3/4| for all m,n. Hence, we have


When n>2m, we have n2(2m+1)2=4m2+4m+1>4m2+4m-3=4vm. Hence, |n2-vm|>3n2/4 and


Finally 1/(2vm)<1/2 since vm>1 when m1. Also, |e-yv|=e-yv-e-yvm<e-ym2. From these observations, we conclude that


Note that this quantity approaches 0 in the limit m.

Let Pm be the arc of the parabola P boundedPlanetmathPlanetmathPlanetmathPlanetmath by the endpointsMathworldPlanetmath of Sm. Together, Sm and Pm form a closed contour which encloses poles of the integrand. Hence, by the residue theoremMathworldPlanetmath , we have


Taking the limit m we obtain


Going back to the beginning of the proof, where the integral on the left hand side was expressed as an integral with respect to u, we obtain


Making a change of variables x=2z,y=-iπτ and tidying up some, we obtain


Because of the initial assumptionPlanetmathPlanetmath about the Fourier series, we only know that this formulaMathworldPlanetmathPlanetmath is valid when τ is purely imaginary with strictly positive imaginary part and z is real and π/2<z<π/2. However, we can use analytic continuation to extend the domain of its validity. On the one hand, the theta function on the right-hand side is analyticPlanetmathPlanetmath for all z and all τ such that τ>0.

On the other hand, I claim that the integral on the left hand side is also an analytic function of z and τ whenever τ>0. To validate this claim, we need to examine the behaviour of the integrand as ui±. The contribution of the denominator is bounded;


for some constant c whenever u=1. The absolute value of the cosine in the numerator is easy to bound:


To bound the remaining term, let us examine the argument of the exponential carefully:


Therefore, if |u|>1+3|τ|/(τ), it will be the case that (τu2)τ(u)2/9, and so


Taken together, the estimates of the last paragraph imply that


when R>1+3|τ|/(τ). If we impose the further conditions

R>180|z|πτ  R2>180|z|πτ  ,

it will be the case that

-πτ(u)2/10  ,

and hence

|i+Ri+cos(2uz)eiπτu2sin(πu)𝑑u|<ci+Ri+e-πτ(u)2/10𝑑u<5cπτRe-πτR2/10  .

Likewise, under the same restrictionPlanetmathPlanetmathPlanetmath on R,

|i-i-Rcos(2uz)eiπτu2sin(πu)𝑑u|<ci+Ri+e-πτ(u)2/10𝑑u<5cπτRe-πτR2/10  .

Since the contour of integration is compactPlanetmathPlanetmath and the integrand is analytic in a neighborhoodMathworldPlanetmathPlanetmath of the contour,


will be an analytic function of z and τ. Suppose that z and τ are restricted to bounded regions of the complex plane and that, furthermore, Imτ is positive and bounded away from zero. Then the inequalities of the last paragraph imply that the integral converges uniformly as R, and hence


is an analytic function of u and z in the domain τ>0.

Thus, by the fundamental theorem of analytic continuation, we may conclude that


throughout this domain.

Finally, integral representations of the remaining three theta functions may be easily obtained from this one by adding the appropriate half-quasiperiods to z.

Title derivation of integral representations of Jacobi ϑ functions
Canonical name DerivationOfIntegralRepresentationsOfJacobivarthetaFunctions
Date of creation 2013-03-22 14:39:54
Last modified on 2013-03-22 14:39:54
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 26
Author rspuzio (6075)
Entry type Derivation
Classification msc 33E05