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8

Special Functions

8.1 Elliptic Integrals and Functions

8.11 Elliptic integrals

8.110

1. Every integral of the form $∫R(x,P(x))dx$, where P(x) is a third- or fourth-degree polynomial, can be reduced to a linear combination of integrals leading to elementary functions and the following three integrals:

$\begin{array}{ccc}\int \frac{\text{d}x}{\sqrt{\left(1-{x}^{2}\right)\left(1-{k}^{2}{x}^{2}\right)}},& \int \frac{\sqrt{\left(1-{k}^{2}{x}^{2}\right)}}{\left(1-{x}^{2}\right)}\text{d}x,& \int \frac{\text{d}x}{\left(1-n{x}^{2}\right)\sqrt{\left(1-{x}^{2}\right)\left(1-{k}^{2}{x}^{2}\right)}},\end{array}$

which are called respectively elliptic integrals of the first, second, and third kind in the Legendre normal form. The results of this reduction for the more frequently encountered ...

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