The theta functions of Jacobi

Peter Lindqvist, rum 1152-SII

These functions are very effective from a computational point of view andthey deserve to be better known. Virtually most calculations of elliptic(=double periodic) functions are based on such rapidly convergent seriesas

θ3(z, q) = 1 + 2q cos 2z + 2q4 cos 4z + 2q9 cos 6z + · · ·

This is one of them. Identities like

(

∞∑

n=−∞

qn2

)4

=

(

∞∑

n=−∞

(−1)nqn2

)4

+

(

∞∑

n=−∞

q(n+ 1

2)2

)4

and∞∑

n=−∞

e−n2πt =1√

t

∞∑

n=−∞

e−n2π/t

abandon. - Although “everything“ about the theta functions can be found inthe literature, this project requires some mathematical ability of the student.(Those that enjoy abstract reasoning instead of solid calculations should notconsider this project.)

References

R. BELLMAN: A Brief introduction to Theta Functions, New York 1961E. WHITTAKER & G. WATSON: A Course of Modern Analysis.