19 Elliptic Integrals Legendre’s Integrals 19.5 Maclaurin and Related Expansions 19.7 Connection Formulas

§19.6 Special Cases

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Keywords:: Legendre’s elliptic integrals, limiting values, special cases
Permalink:: http://dlmf.nist.gov/19.6
See also:: Annotations for Ch.19

§19.6(i) Complete Elliptic Integrals
§19.6(ii) $F\left(\phi,k\right)$
§19.6(iii) $E\left(\phi,k\right)$
§19.6(iv) $\Pi\left(\phi,\alpha^{2},k\right)$
§19.6(v) $R_{C}\left(x,y\right)$

§19.6(i) Complete Elliptic Integrals

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Notes:: For the first line of (19.6.2) put $\alpha=k$ in the first line of (19.25.2) and use the last line of (19.25.1). For the second line of (19.6.2), and also for (19.6.5), use (19.7.8) and (19.6.15). For the first line of (19.6.6) use (19.6.5) and (19.6.2). For more detail as $k^{2}\rightarrow 1-$ see §19.12.
Referenced by:: Figure 19.3.5, Figure 19.3.5, Figure 19.3.5
Permalink:: http://dlmf.nist.gov/19.6.i
See also:: Annotations for §19.6 and Ch.19

19.6.1

$K\left(0\right)=E\left(0\right)={K^{\prime}}\left(1\right)={E^{\prime}}\left(1% \right)=\tfrac{1}{2}\pi,$

$K\left(1\right)={K^{\prime}}\left(0\right)=\infty,$

$E\left(1\right)={E^{\prime}}\left(0\right)=1.$

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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, ${E^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the second kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind and $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind
Permalink:: http://dlmf.nist.gov/19.6.E1
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §19.6(i), §19.6 and Ch.19

19.6.2

$\Pi\left(k^{2},k\right)=E\left(k\right)/{k^{\prime}}^{2},$ $k^{2}<1$ ,

$\Pi\left(-k,k\right)=\tfrac{1}{4}\pi(1+k)^{-1}+\tfrac{1}{2}K\left(k\right),$ $0\leq k^{2}<1$ .

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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, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, : real or complex modulus and $k^{\prime}$ : complementary modulus
Referenced by:: §19.6(i)
Permalink:: http://dlmf.nist.gov/19.6.E2
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §19.6(i), §19.6 and Ch.19

19.6.3 $\Pi\left(\alpha^{2},0\right)=\pi/(2\sqrt{1-\alpha^{2}}),\quad\Pi\left(0,k% \right)=K\left(k\right),$ $-\infty<\alpha^{2}<1$ .

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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, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/19.6.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §19.6(i), §19.6 and Ch.19

19.6.4

$\Pi\left(\alpha^{2},k\right)\to+\infty,$ $\alpha^{2}\to 1-$ ,

$\Pi\left(\alpha^{2},k\right)\to\infty\operatorname{sign}\left(1-\alpha^{2}% \right),$ $k^{2}\to 1-$ .

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Symbols:: $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $\operatorname{sign}\NVar{x}$ : sign of, : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/19.6.E4
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §19.6(i), §19.6 and Ch.19

If $1<\alpha^{2}<\infty$ , then the Cauchy principal value satisfies

19.6.5 $\Pi\left(\alpha^{2},k\right)=K\left(k\right)-\Pi\left(k^{2}/\alpha^{2},k\right),$

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Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.12, Figure 19.3.6, Figure 19.3.6, Figure 19.3.6, §19.6(i), §19.7(iii), §19.8(i)
Permalink:: http://dlmf.nist.gov/19.6.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §19.6(i), §19.6 and Ch.19

and

19.6.6

$\Pi\left(\alpha^{2},0\right)=0,$

$\Pi\left(\alpha^{2},k\right)\to K\left(k\right)-\left(E\left(k\right)/{k^{% \prime}}^{2}\right),$ $\alpha^{2}\to 1+$ ,

$\Pi\left(\alpha^{2},k\right)\to-\infty,$ $k^{2}\rightarrow 1-$ .

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Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: Figure 19.3.6, Figure 19.3.6, Figure 19.3.6, §19.6(i)
Permalink:: http://dlmf.nist.gov/19.6.E6
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §19.6(i), §19.6 and Ch.19

Exact values of $K\left(k\right)$ and $E\left(k\right)$ for various special values of are given in Byrd and Friedman (1971, 111.10 and 111.11) and Cooper et al. (2006).

§19.6(ii) $F\left(\phi,k\right)$

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Keywords:: Legendre’s elliptic integrals, $R_{C}$ -function, inverse Gudermannian function, relation to Legendre’s elliptic integrals, relation to $R_{C}$ -function, relation to inverse Gudermannian function, relations to other functions
Notes:: Use (19.2.4), (19.16.6), and (19.25.5).
Permalink:: http://dlmf.nist.gov/19.6.ii
See also:: Annotations for §19.6 and Ch.19

19.6.7

$F\left(0,k\right)=0,$

$F\left(\phi,0\right)=\phi,$

$F\left(\tfrac{1}{2}\pi,1\right)=\infty,$

$F\left(\tfrac{1}{2}\pi,k\right)=K\left(k\right),$

$\lim_{\phi\to 0}\ifrac{F\left(\phi,k\right)}{\phi}=1.$

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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, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument and : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.6.E7
Encodings:: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png
See also:: Annotations for §19.6(ii), §19.6 and Ch.19

19.6.8 $F\left(\phi,1\right)=(\sin\phi)R_{C}\left(1,{\cos}^{2}\phi\right)={% \operatorname{gd}^{-1}}\left(\phi\right).$

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, ${\operatorname{gd}^{-1}}\left(\NVar{x}\right)$ : inverse Gudermannian function, $\sin\NVar{z}$ : sine function and $\phi$ : real or complex argument
Referenced by:: §19.10(ii), §19.10(ii)
Permalink:: http://dlmf.nist.gov/19.6.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §19.6(ii), §19.6 and Ch.19

For the inverse Gudermannian function ${\operatorname{gd}^{-1}}\left(\phi\right)$ see §4.23(viii). Compare also (19.10.2).

§19.6(iii) $E\left(\phi,k\right)$

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Notes:: Use (19.2.5).
Permalink:: http://dlmf.nist.gov/19.6.iii
See also:: Annotations for §19.6 and Ch.19

19.6.9

$E\left(0,k\right)=0,$

$E\left(\phi,0\right)=\phi,$

$E\left(\tfrac{1}{2}\pi,1\right)=1,$

$E\left(\phi,1\right)=\sin\phi,$

$E\left(\tfrac{1}{2}\pi,k\right)=E\left(k\right).$

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Symbols:: $\pi$ : 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, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument and : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.6.E9
Encodings:: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png
See also:: Annotations for §19.6(iii), §19.6 and Ch.19

19.6.10 $\lim_{\phi\to 0}\ifrac{E\left(\phi,k\right)}{\phi}=1.$

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Symbols:: $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\phi$ : real or complex argument and : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.6.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §19.6(iii), §19.6 and Ch.19

§19.6(iv) $\Pi\left(\phi,\alpha^{2},k\right)$

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Notes:: Byrd and Friedman (1971, 111.01 and 111.04, p. 10) also needs $\alpha\sin\phi<1$ . Start with (19.25.14). For the second equation of (19.6.12) use (19.20.8). For (19.6.13) use (19.16.5) with (19.25.10) and (19.25.11).
Permalink:: http://dlmf.nist.gov/19.6.iv
See also:: Annotations for §19.6 and Ch.19

Circular and hyperbolic cases, including Cauchy principal values, are unified by using $R_{C}\left(x,y\right)$ . Let $c={\csc}^{2}\phi\neq\alpha^{2}$ and $\Delta=\sqrt{1-k^{2}{\sin}^{2}\phi}$ . Then