19 Elliptic Integrals Legendre’s Integrals 19.4 Derivatives and Differential Equations 19.6 Special Cases

§19.5 Maclaurin and Related Expansions

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Keywords:: Appell functions, Jacobi’s, Jacobi’s nome, Legendre’s elliptic integrals, nome, power-series expansion, power-series expansions, relation to Legendre’s elliptic integrals, relations to other functions
Notes:: For (19.5.1)–(19.5.4) put $\sin\phi=1$ and $t=\sqrt{x}$ in (19.2.4)–(19.2.7). Then compare with Erdélyi et al. (1953a, 2.1.3(10) and 2.1.1(2)) in the first three cases, and with Erdélyi et al. (1953a, 5.8.2(5) and 5.7.1(6)) in the fourth case. For (19.5.5) and (19.5.6) see Kneser (1927, (12) and p. 218); Byrd and Friedman (1971, 901.00) is incorrect. (19.5.8) and (19.5.9) follow from Borwein and Borwein (1987, (2.1.13) and (2.3.17), respectively). For (19.5.10) iterate (19.8.12).
Referenced by:: §14.5(v), §19.36(iv), §20.11(iii), §22.5(ii)
Permalink:: http://dlmf.nist.gov/19.5
Addition (effective with 1.1.0):: Equations (19.5.4_1), (19.5.4_2) and (19.5.4_3) were added.
See also:: Annotations for Ch.19

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

19.5.1		$K\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2% }F_{1}}\left({\tfrac{1}{2},\tfrac{1}{2}\atop 1};k^{2}\right),$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer and $k$ : real or complex modulus Source: Carlson (1961b, (2.4)) Referenced by: §19.36(iv), §19.5 Permalink: http://dlmf.nist.gov/19.5.E1 Encodings: TeX, pMML, png See also: Annotations for §19.5 and Ch.19

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

19.5.2		$E\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2% }F_{1}}\left({-\tfrac{1}{2},\tfrac{1}{2}\atop 1};k^{2}\right),$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer and $k$ : real or complex modulus Source: Carlson (1961b, (2.5)) Referenced by: §19.36(iv) Permalink: http://dlmf.nist.gov/19.5.E2 Encodings: TeX, pMML, png See also: Annotations for §19.5 and Ch.19

19.5.3		$D\left(k\right)=\frac{\pi}{4}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{3}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{(m+1)!\;m!}k^{2m}=\frac{\pi}{4}{{% }_{2}F_{1}}\left({\tfrac{3}{2},\tfrac{1}{2}\atop 2};k^{2}\right),$
ⓘ Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $D\left(\NVar{k}\right)$ : complete elliptic integral of Legendre’s type, $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer and $k$ : real or complex modulus Permalink: http://dlmf.nist.gov/19.5.E3 Encodings: TeX, pMML, png See also: Annotations for §19.5 and Ch.19

19.5.4		$\Pi\left(\alpha^{2},k\right)=\frac{\pi}{2}\sum_{n=0}^{\infty}\frac{{\left(% \tfrac{1}{2}\right)_{n}}}{n!}\sum_{m=0}^{n}\frac{{\left(\tfrac{1}{2}\right)_{m% }}}{m!}k^{2m}\alpha^{2n-2m}=\frac{\pi}{2}{F_{1}}\left(\tfrac{1}{2};\tfrac{1}{2% },1;1;k^{2},\alpha^{2}\right),$
ⓘ Symbols: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $!$ : factorial (as in $n!$ ), $m$ : nonnegative integer, $n$ : nonnegative integer, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter Source: Carlson (1961b, (2.10)) Referenced by: §19.5 Permalink: http://dlmf.nist.gov/19.5.E4 Encodings: TeX, pMML, png See also: Annotations for §19.5 and Ch.19

19.5.4_1		$F\left(\phi,k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{% \sin}^{2m+1}\phi}{(2m+1)m!}{{}_{2}F_{1}}\left({m+\tfrac{1}{2},\tfrac{1}{2}% \atop m+\tfrac{3}{2}};{\sin}^{2}{\phi}\right)k^{2m}=\sin\phi\,{F_{1}}\left(% \tfrac{1}{2};\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};{\sin}^{2}\phi,k^{2}{\sin}% ^{2}\phi\right),$
ⓘ Symbols: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $\phi$ : real or complex argument and $k$ : real or complex modulus Source: Carlson (1961b, (2.8)) Referenced by: §19.5, Erratum (V1.1.2) for Additions Permalink: http://dlmf.nist.gov/19.5.E4_1 Encodings: TeX, pMML, png Addition (effective with 1.1.2): This equation was added. See also: Annotations for §19.5 and Ch.19

19.5.4_2		$E\left(\phi,k\right)=\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}\right)_{m}}% {\sin}^{2m+1}\phi}{(2m+1)m!}{{}_{2}F_{1}}\left({m+\tfrac{1}{2},\tfrac{1}{2}% \atop m+\tfrac{3}{2}};{\sin}^{2}{\phi}\right)k^{2m}=\sin\phi\,{F_{1}}\left(% \tfrac{1}{2};\tfrac{1}{2},-\tfrac{1}{2};\tfrac{3}{2};{\sin}^{2}\phi,k^{2}{\sin% }^{2}\phi\right),$
ⓘ Symbols: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $\phi$ : real or complex argument and $k$ : real or complex modulus Source: Carlson (1961b, (2.9)) Referenced by: §19.5, Erratum (V1.1.2) for Additions Permalink: http://dlmf.nist.gov/19.5.E4_2 Encodings: TeX, pMML, png Addition (effective with 1.1.2): This equation was added. See also: Annotations for §19.5 and Ch.19

19.5.4_3		$\Pi\left(\phi,\alpha^{2},k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\sin}^{2m+1}\phi}{(2m+1)m!}{F_{1}}\left(m+\tfrac{1}{2};\tfrac{1}{% 2},1;m+\tfrac{3}{2};{\sin}^{2}\phi,\alpha^{2}{\sin}^{2}\phi\right)k^{2m},$
ⓘ Symbols: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter Source: Carlson (1961b, (2.11)) Referenced by: §19.5, Erratum (V1.1.2) for Additions Permalink: http://dlmf.nist.gov/19.5.E4_3 Encodings: TeX, pMML, png Addition (effective with 1.1.2): This equation was added. See also: Annotations for §19.5 and Ch.19

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		$q=\exp\left(-\pi{K^{\prime}}\left(k\right)/K\left(k\right)\right)=r+8r^{2}+84r% ^{3}+992r^{4}+\cdots,$
$r=\frac{1}{16}k^{2}$ , $0\leq k\leq 1$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\exp\NVar{z}$ : exponential function, $q$ : nome and $k$ : real or complex modulus Referenced by: §19.5 Permalink: http://dlmf.nist.gov/19.5.E5 Encodings: TeX, pMML, png See also: Annotations for §19.5 and Ch.19

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		$K\left(k\right)=\frac{\pi}{2}\left(1+2\sum_{n=1}^{\infty}q^{n^{2}}\right)^{2},$
$\|q\|<1$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $q$ : nome, $n$ : nonnegative integer and $k$ : real or complex modulus Referenced by: §19.5, §19.5 Permalink: http://dlmf.nist.gov/19.5.E8 Encodings: TeX, pMML, png See also: Annotations for §19.5 and Ch.19

19.5.9		$E\left(k\right)=K\left(k\right)+\frac{2\pi^{2}}{K\left(k\right)}\,\frac{\sum_{% n=1}^{\infty}(-1)^{n}n^{2}q^{n^{2}}}{1+2\sum_{n=1}^{\infty}(-1)^{n}q^{n^{2}}},$
$\|q\|<1$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $q$ : nome, $n$ : nonnegative integer and $k$ : real or complex modulus Referenced by: §19.5, §19.5 Permalink: http://dlmf.nist.gov/19.5.E9 Encodings: TeX, pMML, png See also: Annotations for §19.5 and Ch.19

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

19.5.10		$K\left(k\right)=\frac{\pi}{2}\prod_{m=1}^{\infty}(1+k_{m}),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $m$ : nonnegative integer and $k$ : real or complex modulus Referenced by: §19.5 Permalink: http://dlmf.nist.gov/19.5.E10 Encodings: TeX, pMML, png See also: Annotations for §19.5 and Ch.19

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).

19.4 Derivatives and Differential Equations 19.6 Special Cases

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