§19.5 Maclaurin and Related Expansions

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

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

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

 19.5.2 $\mathop{E\/}\nolimits\!\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{% \left(-\tfrac{1}{2}\right)_{m}\left(\tfrac{1}{2}\right)_{m}}{m!\;m!}k^{2m}=% \frac{\pi}{2}\mathop{{{}_{2}F_{1}}\/}\nolimits\!\left(-\tfrac{1}{2},\tfrac{1}{% 2};1;k^{2}\right),$
 19.5.3 $\mathop{D\/}\nolimits\!\left(k\right)=\frac{\pi}{4}\sum_{m=0}^{\infty}\frac{% \left(\tfrac{3}{2}\right)_{m}\left(\tfrac{1}{2}\right)_{m}}{(m+1)!\;m!}k^{2m}=% \frac{\pi}{4}\mathop{{{}_{2}F_{1}}\/}\nolimits\!\left(\tfrac{3}{2},\tfrac{1}{2% };2;k^{2}\right),$
 19.5.4 $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)=\frac{\pi}{2}\sum_{n=0}^{% \infty}\frac{\left(\tfrac{1}{2}\right)_{n}}{n!}\sum_{m=0}^{n}\frac{\left(% \tfrac{1}{2}\right)_{m}}{m!}k^{2m}\alpha^{2n-2m}=\frac{\pi}{2}\mathop{{F_{1}}% \/}\nolimits\!\left(\tfrac{1}{2};\tfrac{1}{2},1;1;k^{2},\alpha^{2}\right),$

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

For Jacobi’s nome $q$:

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

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

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

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

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

An infinite series for $\mathop{\ln\/}\nolimits\mathop{K\/}\nolimits\!\left(k\right)$ is equivalent to the infinite product

 19.5.10 $\mathop{K\/}\nolimits\!\left(k\right)=\frac{\pi}{2}\prod_{m=1}^{\infty}(1+k_{m% }),$

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

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