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

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8 Elliptic integrals

8.1 Elliptic integrals in general

The treatment of the elliptic integrals given in this chapter (and later work on their applications, for example in Chapter 12) owes much to the books by Bowman (1961), Dutta and Debnath (1965), Lawden (1989) and Whittaker and Watson (1927). Detailed references are given as appropriate in what follows.

An integral of the type

∫R⁡(x,y)dx,(8.1)

where y2 is a cubic or quartic polynomial in x, and R denotes a rational function of x and y, is called an elliptic integral1 (though, strictly speaking, it is an elliptic integral only if it cannot be integrated using elementary functions). Such integrals include the ...

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