# §19.5 Maclaurin and Related Expansions

If $|k|<1$ and $|\alpha|<1$, then

 19.5.1 $K\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2% }F_{1}}\left({\tfrac{1}{2},\tfrac{1}{2}\atop 1};k^{2}\right),$

where ${{}_{2}F_{1}}$ is the Gauss hypergeometric function (§§15.1 and 15.2(i)).

 19.5.2 $E\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2% }F_{1}}\left({-\tfrac{1}{2},\tfrac{1}{2}\atop 1};k^{2}\right),$
 19.5.3 $D\left(k\right)=\frac{\pi}{4}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{3}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{(m+1)!\;m!}k^{2m}=\frac{\pi}{4}{{% }_{2}F_{1}}\left({\tfrac{3}{2},\tfrac{1}{2}\atop 2};k^{2}\right),$
 19.5.4 $\Pi\left(\alpha^{2},k\right)=\frac{\pi}{2}\sum_{n=0}^{\infty}\frac{{\left(% \tfrac{1}{2}\right)_{n}}}{n!}\sum_{m=0}^{n}\frac{{\left(\tfrac{1}{2}\right)_{m% }}}{m!}k^{2m}\alpha^{2n-2m}=\frac{\pi}{2}{F_{1}}\left(\tfrac{1}{2};\tfrac{1}{2% },1;1;k^{2},\alpha^{2}\right),$
 19.5.4_1 $F\left(\phi,k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{% \sin}^{2m+1}\phi}{(2m+1)m!}{{}_{2}F_{1}}\left({m+\tfrac{1}{2},\tfrac{1}{2}% \atop m+\tfrac{3}{2}};{\sin}^{2}{\phi}\right)k^{2m}=\sin\phi\,{F_{1}}\left(% \tfrac{1}{2};\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};{\sin}^{2}\phi,k^{2}{\sin}% ^{2}\phi\right),$
 19.5.4_2 $E\left(\phi,k\right)=\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}\right)_{m}}% {\sin}^{2m+1}\phi}{(2m+1)m!}{{}_{2}F_{1}}\left({m+\tfrac{1}{2},\tfrac{1}{2}% \atop m+\tfrac{3}{2}};{\sin}^{2}{\phi}\right)k^{2m}=\sin\phi\,{F_{1}}\left(% \tfrac{1}{2};\tfrac{1}{2},-\tfrac{1}{2};\tfrac{3}{2};{\sin}^{2}\phi,k^{2}{\sin% }^{2}\phi\right),$
 19.5.4_3 $\Pi\left(\phi,\alpha^{2},k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\sin}^{2m+1}\phi}{(2m+1)m!}{F_{1}}\left(m+\tfrac{1}{2};\tfrac{1}{% 2},1;m+\tfrac{3}{2};{\sin}^{2}\phi,\alpha^{2}{\sin}^{2}\phi\right)k^{2m},$

where ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)$ is an Appell function (§16.13).

For Jacobi’s nome $q$:

 19.5.5 $q=\exp\left(-\pi{K^{\prime}}\left(k\right)/K\left(k\right)\right)=r+8r^{2}+84r% ^{3}+992r^{4}+\cdots,$ $r=\frac{1}{16}k^{2}$, $0\leq k\leq 1$.

Also,

 19.5.6 $q=\lambda+2\lambda^{5}+15\lambda^{9}+150\lambda^{13}+1707\lambda^{17}+\cdots,$ $0\leq k\leq 1$, ⓘ Symbols: $q$: nome and $k$: real or complex modulus Referenced by: §19.5, §19.5 Permalink: http://dlmf.nist.gov/19.5.E6 Encodings: TeX, pMML, png See also: Annotations for §19.5 and Ch.19

where

 19.5.7 $\lambda=(1-\sqrt{k^{\prime}})/(2(1+\sqrt{k^{\prime}})).$ ⓘ Symbols: $k^{\prime}$: complementary modulus Permalink: http://dlmf.nist.gov/19.5.E7 Encodings: TeX, pMML, png See also: Annotations for §19.5 and Ch.19

Coefficients of terms up to $\lambda^{49}$ are given in Lee (1990), along with tables of fractional errors in $K\left(k\right)$ and $E\left(k\right)$, $0.1\leq k^{2}\leq 0.9999$, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9).

 19.5.8 $K\left(k\right)=\frac{\pi}{2}\left(1+2\sum_{n=1}^{\infty}q^{n^{2}}\right)^{2},$ $|q|<1$,
 19.5.9 $E\left(k\right)=K\left(k\right)+\frac{2\pi^{2}}{K\left(k\right)}\,\frac{\sum_{% n=1}^{\infty}(-1)^{n}n^{2}q^{n^{2}}}{1+2\sum_{n=1}^{\infty}(-1)^{n}q^{n^{2}}},$ $|q|<1$.

An infinite series for $\ln K\left(k\right)$ is equivalent to the infinite product

 19.5.10 $K\left(k\right)=\frac{\pi}{2}\prod_{m=1}^{\infty}(1+k_{m}),$ ⓘ

where $k_{0}=k$ and

 19.5.11 $k_{m+1}=\frac{1-\sqrt{1-k_{m}^{2}}}{1+\sqrt{1-k_{m}^{2}}},$ $m=0,1,\dots$. ⓘ Symbols: $m$: nonnegative integer and $k$: real or complex modulus Permalink: http://dlmf.nist.gov/19.5.E11 Encodings: TeX, pMML, png See also: Annotations for §19.5 and Ch.19

Series expansions of $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$ are surveyed and improved in Van de Vel (1969), and the case of $F\left(\phi,k\right)$ is summarized in Gautschi (1975, §1.3.2). For series expansions of $\Pi\left(\phi,\alpha^{2},k\right)$ when $|\alpha^{2}|<1$ see Erdélyi et al. (1953b, §13.6(9)). See also Karp et al. (2007).