§19.5 Maclaurin and Related Expansions
If and
, 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),](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E1.png)
where 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),](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E2.png)
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),](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E3.png)
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),](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E4.png)
where is an Appell function
(§16.13).
For Jacobi’s nome :
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,](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E5.png)
,
.
![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,](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E5.png)
![r=\frac{1}{16}k^{2}](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/m29.png)
![0\leq k\leq 1](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/m3.png)
Also,
19.5.6![q=\lambda+2\lambda^{5}+15\lambda^{9}+150\lambda^{{13}}+1707\lambda^{{17}}+\cdots,](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E6.png)
,
![q=\lambda+2\lambda^{5}+15\lambda^{9}+150\lambda^{{13}}+1707\lambda^{{17}}+\cdots,](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E6.png)
![0\leq k\leq 1](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/m3.png)
where
19.5.7
![\lambda=(1-\sqrt{k^{{\prime}}})/(2(1+\sqrt{k^{{\prime}}})).](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E7.png)
Coefficients of terms up to are given in Lee (1990),
along with tables of fractional errors in
and
,
, 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},](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E8.png)
,
![\mathop{K\/}\nolimits\!\left(k\right)=\frac{\pi}{2}\left(1+2\sum _{{n=1}}^{{\infty}}q^{{n^{2}}}\right)^{2},](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E8.png)
![|q|<1](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/m34.png)
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}}}},](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E9.png)
.
![\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}}}},](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E9.png)
![|q|<1](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/m34.png)
An infinite series for 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}),](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E10.png)
where and
19.5.11![k_{{m+1}}=\frac{1-\sqrt{1-k_{m}^{2}}}{1+\sqrt{1-k_{m}^{2}}},](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E11.png)
.
![k_{{m+1}}=\frac{1-\sqrt{1-k_{m}^{2}}}{1+\sqrt{1-k_{m}^{2}}},](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/E11.png)
![m=0,1,\dots](https://webarchive.library.unt.edu/web/20130301173707im_/http://dlmf.nist.gov/19/5/m24.png)