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

§19.5 Maclaurin and Related Expansions

Notes:: For (19.5.1)–(19.5.4) put $\mathop{\sin\/}\nolimits\phi=1$ 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).
Keywords:: Appell functions, Jacobi’s nome, Legendre’s elliptic integrals, nome
Referenced by:: §14.5(v), §19.36(iv), §20.11(iii), §22.5(ii)
Permalink:: http://dlmf.nist.gov/19.5

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

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

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : : factorial, : nonnegative integer and : real or complex modulus
Referenced by:: §19.36(iv), §19.5
Permalink:: http://dlmf.nist.gov/19.5.E1
Encodings:: TeX, pMML, png

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

19.5.2 $\mathop{E\/}\nolimits\!\left(k\right)=\frac{\pi}{2}\sum_{{m=0}}^{{\infty}}% \frac{\left(-\tfrac{1}{2}\right)_{{m}}\left(\tfrac{1}{2}\right)_{{m}}}{m!\;m!}% k^{{2m}}=\frac{\pi}{2}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(-\tfrac{1}{% 2},\tfrac{1}{2};1;k^{2}\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, : : factorial, : nonnegative integer and : real or complex modulus
Referenced by:: §19.36(iv)
Permalink:: http://dlmf.nist.gov/19.5.E2
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{D\/}\nolimits\!\left(k\right)$ : complete elliptic integral of Legendre’s type, : : factorial, : nonnegative integer and : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.5.E3
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{{F_{{1}}}\/}\nolimits\!\left(\alpha;\beta,\beta^{{\prime}};\gamma;x,y\right)$ : Appell function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind, : : factorial, : nonnegative integer, : nonnegative integer, : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.5
Permalink:: http://dlmf.nist.gov/19.5.E4
Encodings:: TeX, pMML, png

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

For Jacobi’s nome :

19.5.5 $q=\mathop{\exp\/}\nolimits\!\left(-\pi\mathop{{K^{{\prime}}}\/}\nolimits\!% \left(k\right)/\mathop{K\/}\nolimits\!\left(k\right)\right)=r+8r^{2}+84r^{3}+9% 92r^{4}+\cdots,$ $r=\frac{1}{16}k^{2}$ , $0\leq k\leq 1$ .

Symbols:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\exp\/}\nolimits z$ : exponential function, : nome and : real or complex modulus
Referenced by:: §19.5
Permalink:: http://dlmf.nist.gov/19.5.E5
Encodings:: TeX, pMML, png

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:: : nome and : 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 $\mathop{K\/}\nolimits\!\left(k\right)=\frac{\pi}{2}\left(1+2\sum_{{n=1}}^{{% \infty}}q^{{n^{2}}}\right)^{2},$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nome, : nonnegative integer and : real or complex modulus
Referenced by:: §19.5, §19.5
Permalink:: http://dlmf.nist.gov/19.5.E8
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, : nome, : nonnegative integer and : real or complex modulus
Referenced by:: §19.5, §19.5
Permalink:: http://dlmf.nist.gov/19.5.E9
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer and : real or complex modulus
Referenced by:: §19.5
Permalink:: http://dlmf.nist.gov/19.5.E10
Encodings:: TeX, pMML, png

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:: : nonnegative integer and : 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).

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