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Theta functions are entire functions whose quotients are elliptic functions.
The four principal theta functions are defined by the following rapidly converging series:
θ1(z) = 2q1/4 sin z – 2q9/4 sin 3z + 2q25/4 sin 5z – . . .
θ2(z) = 2q1/4 cos z – 2q9/4 cos 3z + 2q25/4 cos 5z + . . .
θ3(z) = 1 + 2q cos 2z + 2q4 cos 4z + 2q9 cos 6z + . . .
θ4(z) = 1 – 2q cos 2z + 2q4 cos 4z – 2q9 cos 6z + . . .
where ǀqǀ < 1.
When π is added to the argument z, the functions are multiplied by –1, –1, 1, and 1, respectively. When πτ is added to z, the relation between τ and q being given by the equation q = eπiτ the functions are multiplied by the factors – N, N, N, and –N, respectively, where N = q–1e–2ik. It follows that, for example, the quotient θ1(Z)/θ4(Z) is a meromorphic function that does not change when 2π or irr is added to the argument—that is, this quotient is an elliptic function with periods 2π and πτ.
For the representation of automorphic functions, H. Poincaré constructed theta functions that are a generalization of the theta functions described above, which were introduced by K. Jacobi (Jacobi’s notation was somewhat different).