19 Elliptic Integrals Legendre’s Integrals 19.6 Special Cases 19.8 Quadratic Transformations

§19.7 Connection Formulas

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Keywords:: Legendre’s elliptic integrals, connection formulas
Permalink:: http://dlmf.nist.gov/19.7
See also:: Annotations for Ch.19

§19.7(i) Complete Integrals of the First and Second Kinds

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Notes:: Three proofs of (19.7.1) are given in Duren (1991). To prove it from (19.21.1) put $z+1=1/k^{2}$ , use homogeneity, and apply the penultimate equation in (19.25.1) twice.
Referenced by:: Erratum (V1.2.0) for Subsection 19.7(i)
Permalink:: http://dlmf.nist.gov/19.7.i
Correction (effective with 1.2.0):: Just above (19.7.3) the requirement that $\Re{k}>0$ was added.
Suggested 2024-01-12 by Alex Barnett
See also:: Annotations for §19.7 and Ch.19

Legendre’s Relation

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See also:: Annotations for §19.7(i), §19.7 and Ch.19

19.7.1		$E\left(k\right){K^{\prime}}\left(k\right)+{E^{\prime}}\left(k\right)K\left(k% \right)-K\left(k\right){K^{\prime}}\left(k\right)=\tfrac{1}{2}\pi.$
ⓘ 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, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind and $k$ : real or complex modulus Referenced by: §19.21(i), §19.35(i), §19.7(i) Permalink: http://dlmf.nist.gov/19.7.E1 Encodings: TeX, pMML, png See also: Annotations for §19.7(i), §19.7(i), §19.7 and Ch.19

Also,

19.7.2	$\displaystyle K\left(ik/k^{\prime}\right)$	$\displaystyle=k^{\prime}K\left(k\right),$
	$\displaystyle K\left(-ik^{\prime}/k\right)$	$\displaystyle=kK\left(k^{\prime}\right),$
	$\displaystyle E\left(ik/k^{\prime}\right)$	$\displaystyle=(1/k^{\prime})E\left(k\right),$
	$\displaystyle E\left(-ik^{\prime}/k\right)$	$\displaystyle=(1/k)E\left(k^{\prime}\right).$
ⓘ 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, $\mathrm{i}$ : imaginary unit, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus Referenced by: §19.15, Erratum (V1.0.17) for Equation (19.7.2) Permalink: http://dlmf.nist.gov/19.7.E2 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png Clarification (effective with 1.0.17): The argument $k^{\prime}/ik$ has been replaced with $-ik^{\prime}/k$ twice to clarify the meaning. See also: Annotations for §19.7(i), §19.7(i), §19.7 and Ch.19

If $\Re{k}>0$ then

19.7.3	$\displaystyle K\left(1/k\right)$	$\displaystyle=k(K\left(k\right)\mp\mathrm{i}K\left(k^{\prime}\right)),$
	$\displaystyle K\left(1/k^{\prime}\right)$	$\displaystyle=k^{\prime}(K\left(k^{\prime}\right)\pm\mathrm{i}K\left(k\right)),$
	$\displaystyle E\left(1/k\right)$	$\displaystyle=(1/k)\left(E\left(k\right)\pm\mathrm{i}E\left(k^{\prime}\right)-% {k^{\prime}}^{2}K\left(k\right)\mp\mathrm{i}k^{2}K\left(k^{\prime}\right)% \right),$
	$\displaystyle E\left(1/k^{\prime}\right)$	$\displaystyle=(1/k^{\prime})\left(E\left(k^{\prime}\right)\mp\mathrm{i}E\left(% k\right)-k^{2}K\left(k^{\prime}\right)\pm\mathrm{i}{k^{\prime}}^{2}K\left(k% \right)\right),$
ⓘ 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, $\mathrm{i}$ : imaginary unit, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus Referenced by: §19.7(i), Erratum (V1.2.0) for Subsection 19.7(i) Permalink: http://dlmf.nist.gov/19.7.E3 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §19.7(i), §19.7(i), §19.7 and Ch.19

where upper signs apply if $\Im k^{2}>0$ and lower signs if $\Im k^{2}<0$ . This dichotomy of signs (missing in several references) is due to Fettis (1970).

§19.7(ii) Change of Modulus and Amplitude

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Keywords:: Legendre’s elliptic integrals, change of amplitude, change of modulus
Notes:: See the penultimate paragraph in §19.25(i).
Referenced by:: §19.25(i), Erratum (V1.0.7) for Other
Permalink:: http://dlmf.nist.gov/19.7.ii
Addition (effective with 1.1.5):: The text “See also (19.2.10).” was added.
Addition (effective with 1.0.7):: The sentence following (19.7.4) has been added.
Suggested 2013-10-21 by Thomas Coffee
See also:: Annotations for §19.7 and Ch.19

Reciprocal-Modulus Transformation

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Keywords:: Legendre’s elliptic integrals, reciprocal-modulus transformation
See also:: Annotations for §19.7(ii), §19.7 and Ch.19

19.7.4		$\displaystyle F\left(\phi,k_{1}\right)$	$\displaystyle=kF\left(\beta,k\right),$
		$\displaystyle E\left(\phi,k_{1}\right)$	$\displaystyle=(E\left(\beta,k\right)-{k^{\prime}}^{2}F\left(\beta,k\right))/k,$
		$\displaystyle\Pi\left(\phi,\alpha^{2},k_{1}\right)$	$\displaystyle=k\Pi\left(\beta,k^{2}\alpha^{2},k\right),$
	$k_{1}=1/k$ , $\sin\beta=k_{1}\sin\phi\leq 1$ .
ⓘ Defines: $k_{1}$ : change of variable (locally) and $\beta$ : change of variable (locally) Symbols: $F\left(\NVar{\phi},\NVar{k}\right)$ : 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, $\phi$ : real or complex argument, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter Referenced by: §19.15, Figure 19.3.6, Figure 19.3.6, Figure 19.3.6, §19.7(ii) Permalink: http://dlmf.nist.gov/19.7.E4 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.7(ii), §19.7(ii), §19.7 and Ch.19

Provided the functions in these identities are correctly analytically continued in the complex $\beta$ -plane, then the identities will also hold in the complex $\beta$ -plane.

Imaginary-Modulus Transformation

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Keywords:: Legendre’s elliptic integrals, imaginary-modulus transformations
See also:: Annotations for §19.7(ii), §19.7 and Ch.19

19.7.5	$\displaystyle F\left(\phi,ik\right)$	$\displaystyle=\kappa^{\prime}F\left(\theta,\kappa\right),$
	$\displaystyle E\left(\phi,ik\right)$	$\displaystyle=(1/\kappa^{\prime})\left(E\left(\theta,\kappa\right)-\kappa^{2}% \(\sin\theta\cos\theta)\(1-\kappa^{2}{\sin}^{2}\theta)^{-\ifrac{1}{2}}\right),$
	$\displaystyle\Pi\left(\phi,\alpha^{2},ik\right)$	$\displaystyle=(\kappa^{\prime}/\alpha_{1}^{2})\left(\kappa^{2}F\left(\theta,% \kappa\right)+{\kappa^{\prime}}^{2}\alpha^{2}\Pi\left(\theta,\alpha_{1}^{2},% \kappa\right)\right),$
ⓘ 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, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\kappa$ : modulus, change of variable, $\kappa^{\prime}$ : complementary modulus, change of variable, $\theta$ : change of variable and $\alpha_{1}$ : change of variable Permalink: http://dlmf.nist.gov/19.7.E5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.7(ii), §19.7(ii), §19.7 and Ch.19

where

19.7.6	$\displaystyle\kappa$	$\displaystyle=\frac{k}{\sqrt{1+k^{2}}},$
	$\displaystyle\kappa^{\prime}$	$\displaystyle=\frac{1}{\sqrt{1+k^{2}}},$
	$\displaystyle\sin\theta$	$\displaystyle=\frac{\sqrt{1+k^{2}}\sin\phi}{\sqrt{1+k^{2}{\sin}^{2}\phi}},$
	$\displaystyle\alpha_{1}^{2}$	$\displaystyle=\frac{\alpha^{2}+k^{2}}{1+k^{2}}.$
ⓘ Defines: $\kappa$ : modulus, change of variable (locally), $\kappa^{\prime}$ : complementary modulus, change of variable (locally), $\theta$ : change of variable (locally) and $\alpha_{1}$ : change of variable (locally) Symbols: $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter Permalink: http://dlmf.nist.gov/19.7.E6 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §19.7(ii), §19.7(ii), §19.7 and Ch.19

Imaginary-Argument Transformation

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Keywords:: Legendre’s elliptic integrals, imaginary-argument transformations
See also:: Annotations for §19.7(ii), §19.7 and Ch.19

With $\sinh\phi=\tan\psi$ ,

19.7.7	$\displaystyle F\left(i\phi,k\right)$	$\displaystyle=iF\left(\psi,k^{\prime}\right),$
	$\displaystyle E\left(i\phi,k\right)$	$\displaystyle=i\left(F\left(\psi,k^{\prime}\right)-E\left(\psi,k^{\prime}% \right)+(\tan\psi)\sqrt{1-{k^{\prime}}^{2}{\sin}^{2}\psi}\right),$
	$\displaystyle\Pi\left(i\phi,\alpha^{2},k\right)$	$\displaystyle=i\left(F\left(\psi,k^{\prime}\right)-\alpha^{2}\Pi\left(\psi,1-% \alpha^{2},k^{\prime}\right)\right)/{(1-\alpha^{2})}.$
ⓘ Symbols: $F\left(\NVar{\phi},\NVar{k}\right)$ : 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, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus, $\alpha^{2}$ : real or complex parameter and $\psi$ : change of variable Referenced by: §19.15 Permalink: http://dlmf.nist.gov/19.7.E7 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.7(ii), §19.7(ii), §19.7 and Ch.19

For two further transformations of this type see Erdélyi et al. (1953b, p. 316).

§19.7(iii) Change of Parameter of $\Pi\left(\phi,\alpha^{2},k\right)$

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Keywords:: Legendre’s elliptic integrals, Legendre’s relation, change of parameter, circular cases, hyperbolic cases
Notes:: All three relations follow from the change of parameter for the symmetric integral of the third kind; see §19.21(iii) and (19.25.14).
Referenced by:: §19.15, §19.25(i), §19.6(iv)
Permalink:: http://dlmf.nist.gov/19.7.iii
See also:: Annotations for §19.7 and Ch.19

There are three relations connecting $\Pi\left(\phi,\alpha^{2},k\right)$ and $\Pi\left(\phi,\omega^{2},k\right)$ , where $\omega^{2}$ is a rational function of $\alpha^{2}$ . If $k^{2}$ and $\alpha^{2}$ are real, then both integrals are circular cases or both are hyperbolic cases (see §19.2(ii)).

The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of $\Pi\left(\phi,\alpha^{2},k\right)$ when $\alpha^{2}>{\csc}^{2}\phi$ (see (19.6.5) for the complete case). Let $c={\csc}^{2}\phi\neq\alpha^{2}$ . Then

19.7.8		$\Pi\left(\phi,\alpha^{2},k\right)+\Pi\left(\phi,\omega^{2},k\right)=F\left(% \phi,k\right)+\sqrt{c}R_{C}\left((c-1)(c-k^{2}),(c-\alpha^{2})(c-\omega^{2})% \right),$
$\alpha^{2}\omega^{2}=k^{2}$ .
ⓘ Defines: $\omega^{2}$ : change of variable (locally) Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first 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, $\alpha^{2}$ : real or complex parameter and $c$ : change of variable Referenced by: §19.25(i), §19.6(i) Permalink: http://dlmf.nist.gov/19.7.E8 Encodings: TeX, pMML, png See also: Annotations for §19.7(iii), §19.7 and Ch.19

Since $k^{2}\leq c$ we have $\alpha^{2}\omega^{2}\leq c$ ; hence $\alpha^{2}>c$ implies $\omega^{2}<1\leq c$ .

The second relation maps each hyperbolic region onto itself and each circular region onto the other:

19.7.9		$(k^{2}-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)+(k^{2}-\omega^{2})\Pi\left% (\phi,\omega^{2},k\right)=k^{2}F\left(\phi,k\right)-\alpha^{2}\omega^{2}\sqrt{% c-1}R_{C}\left(c(c-k^{2}),(c-\alpha^{2})(c-\omega^{2})\right),$
$(1-\alpha^{2})(1-\omega^{2})=1-k^{2}$ .
ⓘ Defines: $\omega^{2}$ : change of variable (locally) Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first 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, $\alpha^{2}$ : real or complex parameter and $c$ : change of variable Permalink: http://dlmf.nist.gov/19.7.E9 Encodings: TeX, pMML, png See also: Annotations for §19.7(iii), §19.7 and Ch.19

The third relation (missing from the literature of Legendre’s integrals) maps each circular region onto the other and each hyperbolic region onto the other:

19.7.10		$(1-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)+(1-\omega^{2})\Pi\left(\phi,% \omega^{2},k\right)=F\left(\phi,k\right)+(1-\alpha^{2}-\omega^{2})\sqrt{c-k^{2% }}\*R_{C}\left(c(c-1),(c-\alpha^{2})(c-\omega^{2})\right),$
$(k^{2}-\alpha^{2})(k^{2}-\omega^{2})=k^{2}(k^{2}-1)$ .
ⓘ Defines: $\omega^{2}$ : change of variable (locally) Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first 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, $\alpha^{2}$ : real or complex parameter and $c$ : change of variable Permalink: http://dlmf.nist.gov/19.7.E10 Encodings: TeX, pMML, png See also: Annotations for §19.7(iii), §19.7 and Ch.19

19.6 Special Cases 19.8 Quadratic Transformations

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§19.7 Connection Formulas

Contents

§19.7(i) Complete Integrals of the First and Second Kinds

Legendre’s Relation

§19.7(ii) Change of Modulus and Amplitude

Reciprocal-Modulus Transformation

Imaginary-Modulus Transformation

Imaginary-Argument Transformation

§19.7(iii) Change of Parameter of $\Pi\left(\phi,\alpha^{2},k\right)$

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§19.7 Connection Formulas

Contents

§19.7(i) Complete Integrals of the First and Second Kinds

Legendre’s Relation

§19.7(ii) Change of Modulus and Amplitude

Reciprocal-Modulus Transformation

Imaginary-Modulus Transformation

Imaginary-Argument Transformation

§19.7(iii) Change of Parameter of Π⁡(ϕ,α2,k)

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§19.7(iii) Change of Parameter of $\Pi\left(\phi,\alpha^{2},k\right)$