§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	$\displaystyle K\left(0\right)$	$\displaystyle=E\left(0\right)={K^{\prime}}\left(1\right)={E^{\prime}}\left(1% \right)=\tfrac{1}{2}\pi,$
	$\displaystyle K\left(1\right)$	$\displaystyle={K^{\prime}}\left(0\right)=\infty,$
	$\displaystyle E\left(1\right)$	$\displaystyle={E^{\prime}}\left(0\right)=1.$
ⓘ 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, ${E^{\prime}}\left(\NVar{k}\right)$ : 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		$\displaystyle\Pi\left(k^{2},k\right)$	$\displaystyle=E\left(k\right)/{k^{\prime}}^{2},$
	$k^{2}<1$ ,
		$\displaystyle\Pi\left(-k,k\right)$	$\displaystyle=\tfrac{1}{4}\pi(1+k)^{-1}+\tfrac{1}{2}K\left(k\right),$
	$0\leq k^{2}<1$ .
ⓘ 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, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $k$ : 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		$\Pi\left(\alpha^{2},0\right)=\pi/(2\sqrt{1-\alpha^{2}}),\quad\Pi\left(0,k% \right)=K\left(k\right),$
$-\infty<\alpha^{2}<1$ .
ⓘ 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, $k$ : 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		$\displaystyle\Pi\left(\alpha^{2},k\right)$	$\displaystyle\to+\infty,$
	$\alpha^{2}\to 1-$ ,
		$\displaystyle\Pi\left(\alpha^{2},k\right)$	$\displaystyle\to\infty\operatorname{sign}\left(1-\alpha^{2}\right),$
	$k^{2}\to 1-$ .
ⓘ Symbols: $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $\operatorname{sign}\NVar{x}$ : sign of, $k$ : 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		$\Pi\left(\alpha^{2},k\right)=K\left(k\right)-\Pi\left(k^{2}/\alpha^{2},k\right),$
ⓘ 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, $k$ : 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		$\displaystyle\Pi\left(\alpha^{2},0\right)$	$\displaystyle=0,$
		$\displaystyle\Pi\left(\alpha^{2},k\right)$	$\displaystyle\to K\left(k\right)-\left(E\left(k\right)/{k^{\prime}}^{2}\right),$
	$\alpha^{2}\to 1+$ ,
		$\displaystyle\Pi\left(\alpha^{2},k\right)$	$\displaystyle\to-\infty,$
	$k^{2}\rightarrow 1-$ .
ⓘ 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, $k$ : 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 $k$ 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	$\displaystyle F\left(0,k\right)$	$\displaystyle=0,$
	$\displaystyle F\left(\phi,0\right)$	$\displaystyle=\phi,$
	$\displaystyle F\left(\tfrac{1}{2}\pi,1\right)$	$\displaystyle=\infty,$
	$\displaystyle F\left(\tfrac{1}{2}\pi,k\right)$	$\displaystyle=K\left(k\right),$
	$\displaystyle\lim_{\phi\to 0}\ifrac{F\left(\phi,k\right)}{\phi}$	$\displaystyle=1.$
ⓘ 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, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument and $k$ : 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		$F\left(\phi,1\right)=(\sin\phi)R_{C}\left(1,{\cos}^{2}\phi\right)={% \operatorname{gd}^{-1}}\left(\phi\right).$
ⓘ 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	$\displaystyle E\left(0,k\right)$	$\displaystyle=0,$
	$\displaystyle E\left(\phi,0\right)$	$\displaystyle=\phi,$
	$\displaystyle E\left(\tfrac{1}{2}\pi,1\right)$	$\displaystyle=1,$
	$\displaystyle E\left(\phi,1\right)$	$\displaystyle=\sin\phi,$
	$\displaystyle E\left(\tfrac{1}{2}\pi,k\right)$	$\displaystyle=E\left(k\right).$
ⓘ 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 $k$ : 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		$\lim_{\phi\to 0}\ifrac{E\left(\phi,k\right)}{\phi}=1.$
ⓘ Symbols: $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\phi$ : real or complex argument and $k$ : 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

19.6.11	$\displaystyle\Pi\left(0,\alpha^{2},k\right)$	$\displaystyle=0,$
	$\displaystyle\Pi\left(\phi,0,0\right)$	$\displaystyle=\phi,$
	$\displaystyle\Pi\left(\phi,1,0\right)$	$\displaystyle=\tan\phi.$
ⓘ Symbols: $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\tan\NVar{z}$ : tangent function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter Permalink: http://dlmf.nist.gov/19.6.E11 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.6(iv), §19.6 and Ch.19

19.6.12	$\displaystyle\Pi\left(\phi,\alpha^{2},0\right)$	$\displaystyle=R_{C}\left(c-1,c-\alpha^{2}\right),$
	$\displaystyle\Pi\left(\phi,\alpha^{2},1\right)$	$\displaystyle=\frac{1}{1-\alpha^{2}}\left(R_{C}\left(c,c-1\right)-\alpha^{2}R_% {C}\left(c,c-\alpha^{2}\right)\right),$
	$\displaystyle\Pi\left(\phi,1,1\right)$	$\displaystyle=\tfrac{1}{2}(R_{C}\left(c,c-1\right)+\sqrt{c}(c-1)^{-1}).$
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument and $\alpha^{2}$ : real or complex parameter Referenced by: §19.6(iv), §19.9(i), §19.9(ii) Permalink: http://dlmf.nist.gov/19.6.E12 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.6(iv), §19.6 and Ch.19

19.6.13	$\displaystyle\Pi\left(\phi,0,k\right)$	$\displaystyle=F\left(\phi,k\right),$
	$\displaystyle\Pi\left(\phi,k^{2},k\right)$	$\displaystyle=\frac{1}{{k^{\prime}}^{2}}\left(E\left(\phi,k\right)-\frac{k^{2}% }{\Delta}\sin\phi\cos\phi\right),$
	$\displaystyle\Pi\left(\phi,1,k\right)$	$\displaystyle=F\left(\phi,k\right)-\frac{1}{{k^{\prime}}^{2}}(E\left(\phi,k% \right)-\Delta\tan\phi).$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\Delta$ Referenced by: §19.36(ii), §19.6(iv) Permalink: http://dlmf.nist.gov/19.6.E13 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.6(iv), §19.6 and Ch.19

19.6.14	$\displaystyle\Pi\left(\tfrac{1}{2}\pi,\alpha^{2},k\right)$	$\displaystyle=\Pi\left(\alpha^{2},k\right),$
19.6.14	$\displaystyle\lim_{\phi\to 0}\ifrac{\Pi\left(\phi,\alpha^{2},k\right)}{\phi}$	$\displaystyle=1.$
ⓘ Symbols: $\pi$ : 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, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter Permalink: http://dlmf.nist.gov/19.6.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.6(iv), §19.6 and Ch.19

For the Cauchy principal value of $\Pi\left(\phi,\alpha^{2},k\right)$ when $\alpha^{2}>c$ , see §19.7(iii).

§19.6(v) $R_{C}\left(x,y\right)$

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Keywords:: $R_{C}$ -function, limiting values, special values
Permalink:: http://dlmf.nist.gov/19.6.v
See also:: Annotations for §19.6 and Ch.19

19.6.15		$\displaystyle R_{C}\left(x,x\right)$	$\displaystyle=x^{-1/2},$
		$\displaystyle R_{C}\left(\lambda x,\lambda y\right)$	$\displaystyle=\lambda^{-1/2}R_{C}\left(x,y\right),$
		$\displaystyle R_{C}\left(x,y\right)$	$\displaystyle\to+\infty,$
	$y\to 0+$ or $y\to 0-$ , $x>0$ ,
		$\displaystyle R_{C}\left(0,y\right)$	$\displaystyle=\tfrac{1}{2}\pi y^{-1/2},$
	$\|\operatorname{ph}y\|<\pi$ ,
		$\displaystyle R_{C}\left(0,y\right)$	$\displaystyle=0,$
	$y<0$ .
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\pi$ : the ratio of the circumference of a circle to its diameter and $\operatorname{ph}$ : phase Referenced by: §19.2(iv), §19.2(iv), §19.20(iii), §19.6(i), §19.9(i) Permalink: http://dlmf.nist.gov/19.6.E15 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §19.6(v), §19.6 and Ch.19

§19.6 Special Cases

Contents

§19.6(i) Complete Elliptic Integrals

§19.6(ii) F⁡(ϕ,k)

§19.6(iii) E⁡(ϕ,k)

§19.6(iv) Π⁡(ϕ,α2,k)

§19.6(v) RC⁡(x,y)

§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) $R_{C}\left(x,y\right)$