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Elliptic Functions by W. F. Eberlein, J. V. Armitage

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4 Theta functions

In this chapter, we define and prove the basic properties of the theta functions, first studied in depth by Jacobi (1829, 1838). As in Chapter 2, we include a summary of the main results, for easy reference, at the end of the chapter.

4.1 Genesis of the theta functions

We remarked earlier that, in some respects, the limiting case k = 1 (the hyperbolic functions) is a better guie to the behaviour of the Jacobian elliptic functions than is the case k = 0 (the trigonometric functions). Recall that

sn⁡(u,1)=tanh⁡u=sinh⁡u/cosh⁡u

and

cn⁡(u,1)=dn⁡(u,1)=sec⁡hu=1/cosh⁡u.

Now

(4.1)

(For an elementary proof ...

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