19 Elliptic Integrals Legendre’s Integrals 19.11 Addition Theorems 19.13 Integrals of Elliptic Integrals

§19.12 Asymptotic Approximations

Notes:: For (19.12.1) and (19.12.2) see Cayley (1895, p. 54) and Cazenave (1969, pp. 165–169). For (19.12.4) and (19.12.5) use (19.25.2), (19.27.13), and (19.6.5). For (19.12.6) and (19.12.7) see Carlson and Gustafson (1994, (22),(24)).
Keywords:: Legendre’s elliptic integrals, $\mathop{R_{C}\/}\nolimits$ -function
Referenced by:: §19.6(i), §22.3(ii), §22.5(ii), Other Changes, Other Changes, Other Changes
Permalink:: http://dlmf.nist.gov/19.12
Addition and Clarification (effective with 1.0.5):: The reference to Karp and Sitnik (2007) has been added in the middle of this section. Also, the definition of was further clarified (made explicit) after (19.12.3).
Reported 2012-04-02
Clarification (effective with 1.0.3):: The definition of was clarified after (19.12.3).
Reported 2011-08-21

With $\mathop{\psi\/}\nolimits\!\left(x\right)$ denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of $\mathop{K\/}\nolimits\!\left(k\right)$ and $\mathop{E\/}\nolimits\!\left(k\right)$ near the singularity at is given by the following convergent series:

19.12.1 $\mathop{K\/}\nolimits\!\left(k\right)=\sum_{{m=0}}^{{\infty}}\frac{\left(% \tfrac{1}{2}\right)_{{m}}\left(\tfrac{1}{2}\right)_{{m}}}{m!\;m!}{k^{{\prime}}% }^{{2m}}\left(\mathop{\ln\/}\nolimits\left(\frac{1}{k^{{\prime}}}\right)+d(m)% \right),$ $0<|k^{{\prime}}|<1$ ,

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : nonnegative integer, : real or complex modulus, $k^{{\prime}}$ : complementary modulus and : function
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E1
Encodings:: TeX, pMML, png

19.12.2 $\mathop{E\/}\nolimits\!\left(k\right)=1+\frac{1}{2}\sum_{{m=0}}^{{\infty}}% \frac{\left(\tfrac{1}{2}\right)_{{m}}\left(\tfrac{3}{2}\right)_{{m}}}{\left(2% \right)_{{m}}m!}{k^{{\prime}}}^{{2m+2}}\*\left(\mathop{\ln\/}\nolimits\left(% \frac{1}{k^{{\prime}}}\right)+d(m)-\frac{1}{(2m+1)(2m+2)}\right),$ $|k^{{\prime}}|<1$ ,

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, : : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : nonnegative integer, : real or complex modulus, $k^{{\prime}}$ : complementary modulus and : function
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E2
Encodings:: TeX, pMML, png

where

19.12.3

$d(m)=\mathop{\psi\/}\nolimits\!\left(1+m\right)-\mathop{\psi\/}\nolimits\!% \left(\tfrac{1}{2}+m\right),$

$d(m+1)=d(m)-\frac{2}{(2m+1)(2m+2)}$ , $m=0,1,\dots$ ,

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, : nonnegative integer and : function
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E3
Encodings:: TeX, TeX, pMML, pMML, png, png

with $d(0)=2\mathop{\ln\/}\nolimits 2$ .

For the asymptotic behavior of $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ and $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ as $\phi\to\tfrac{1}{2}\pi-$ and $k\to 1-$ see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007).

As $k^{2}\to 1-$

19.12.4 $(1-\alpha^{2})\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)=\left(\mathop% {\ln\/}\nolimits\frac{4}{k^{{\prime}}}\right)\left(1+\mathop{O\/}\nolimits\!% \left({k^{{\prime}}}^{2}\right)\right)-\alpha^{2}\mathop{R_{C}\/}\nolimits\!% \left(1,1-\alpha^{2}\right),$ $-\infty<\alpha^{2}<1$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real or complex modulus, $k^{{\prime}}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E4
Encodings:: TeX, pMML, png

19.12.5 $(1-\alpha^{2})\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)=\left(\mathop% {\ln\/}\nolimits\left(\frac{4}{k^{{\prime}}}\right)-\mathop{R_{C}\/}\nolimits% \!\left(1,1-\alpha^{{-2}}\right)\right)\*\left(1+\mathop{O\/}\nolimits\!\left(% {k^{{\prime}}}^{2}\right)\right),$ $1<\alpha^{2}<\infty$ .

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : real or complex modulus, $k^{{\prime}}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E5
Encodings:: TeX, pMML, png

Asymptotic approximations for $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ , with different variables, are given in Karp et al. (2007). They are useful primarily when $\ifrac{(1-k)}{(1-\mathop{\sin\/}\nolimits\phi)}$ is either small or large compared with 1.

If $x\geq 0$ and , then

19.12.6 $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)=\frac{\pi}{2\sqrt{y}}-\frac{\sqrt{% x}}{y}\left(1+\mathop{O\/}\nolimits\!\left(\sqrt{\frac{x}{y}}\right)\right),$ $x/y\to 0,$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding and $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E6
Encodings:: TeX, pMML, png

19.12.7 $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)=\frac{1}{2\sqrt{x}}\left(\left(1+% \frac{y}{2x}\right)\mathop{\ln\/}\nolimits\left(\frac{4x}{y}\right)-\frac{y}{2% x}\right)\*(1+\mathop{O\/}\nolimits\!\left(y^{2}/x^{2}\right)),$ $y/x\to 0$ .

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function
Referenced by:: §19.12, §19.2(iv)
Permalink:: http://dlmf.nist.gov/19.12.E7
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 19.11 Addition Theorems 19.13 Integrals of Elliptic Integrals