Table of Integrals, Series, and Products, 8th Edition

5

Indefinite Integrals of Special Functions

5.1 Elliptic Integrals and Functions

Notation: $k^{\prime }=\sqrt{1-k^2}\left(\text{cf}\text{. 8}\text{.1}\right).$

5.11 Complete elliptic integrals

5.111

1. 

${\displaystyle \mathit{\int }K\left(k\right)k^{2p+3}}\text{d}k=\frac{1}{{\left(2p+3\right)}^2}\left\{4{\left(p+1\right)}^2{\displaystyle \int K\left(k\right)k^{2p+1}\text{d}k}+k^{2p+2}\left[E\left(k\right)-\left(2p+3\right)K\left(k\right){k^{\prime }}^2\right]\right\}$

  BY (610.04)

2. 

$\begin{array}{ll}{{\displaystyle \int }}^{\text{}}E(k)k^{2p+3}dk=\hfill & \frac{1}{4p^2+16p+15}\{4{(p+1)}^2{{\displaystyle \int }}^{\text{}}E(k)k^{2p+1}dk\hfill \\ -E(k)k^{2p+2}[(2p+3)k^{\prime }{2}^{}-2]-k^{2p+2}{k}^{\prime }{2}^{}K(k)\}\hfill \end{array}$

  BY (611.04)

5.112

1. 

${\displaystyle \int K\left(k\right)}\text{d}k=\frac{\text{π}k}{2}\left[1+{\displaystyle \sum _{j=1}^{\infty }\frac{{\left[\left(2j\right)!\right]}^2{k}^{2j}}{\left(2j+1\right)2^{4j}{\left(j!\right)}^4}}\right]$

  BY (610.00)

2.⁶ 

${\displaystyle \int E\left(k\right)}\text{d}k=\frac{\text{π}k}{2}[1-{\displaystyle \sum _{j=1}^{\infty }\frac{{[(}^{}}{}}$