5

# Indefinite Integrals of Special Functions

## 5.1 Elliptic Integrals and Functions

Notation: ### 5.11 Complete elliptic integrals

5.111

1.

$\mathit{\int }K\left(k\right){k}^{2p+3}\text{d}k=\frac{1}{{\left(2p+3\right)}^{2}}\left\{4{\left(p+1\right)}^{2}\int K\left(k\right){k}^{2p+1}\text{d}k+{k}^{2p+2}\left[E\left(k\right)-\left(2p+3\right)K\left(k\right){{k}^{\prime }}^{2}\right]\right\}$ BY (610.04)

2.

$\begin{array}{ll}{\int }^{\text{​}}E\left(k\right){k}^{2p+3}dk=\hfill & \frac{1}{4{p}^{2}+16p+15}\left\{4{\left(p+1\right)}^{2}{\int }^{\text{​}}E\left(k\right){k}^{2p+1}dk\hfill \\ -E\left(k\right){k}^{2p+2}\left[\left(2p+3\right){k}^{\prime }{2}^{}-2\right]-{k}^{2p+2}{k}^{\prime }{2}^{}K\left(k\right)\right\}\hfill \end{array}$ BY (611.04)

5.112

1.

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

BY (610.00)

2.6

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

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