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.

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

and

Now

- (4.1)

(For an elementary proof ...

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