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

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5 The Jacobian elliptic functions for complex k

5.1 Extension to arbitrary τ with Im(τ) > 0

Hitherto we have supposed (see the Summary, Section 4.9) that 0 < k < 1, from which it follows that τ = it(t > 0). In that case, our basic formulae expressing the Jacobian elliptic functions in terms of the parameter τ are (see Sections 4.3 and 4.4 and Chapter 2):

k1/2=θ2/θ3,(5.1)

k′1/2=θ4/θ3,(5.2)

2K/π=θ32,(5.3)

(5.4)

(5.5a)

(5.5b)

(5.5c)

Now let τ be a complex number, not necessarily ...

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