19 Elliptic Integrals Legendre’s Integrals 19.10 Relations to Other Functions 19.12 Asymptotic Approximations

§19.11 Addition Theorems

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Keywords:: Legendre’s elliptic integrals, addition theorem
Permalink:: http://dlmf.nist.gov/19.11
See also:: Annotations for Ch.19

§19.11(i) General Formulas
§19.11(ii) Case $\psi=\pi/2$
§19.11(iii) Duplication Formulas

§19.11(i) General Formulas

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Notes:: See Byerly (1888, pp. 243–245, 256–258), Edwards (1954, v. 2, pp. 511–513), Cazenave (1969, pp. 83–85), Cayley (1961, pp. 103-106). (19.11.5) can be derived from (19.26.9), (19.25.26), and (19.11.1).
Referenced by:: §19.11(iii), Erratum (V1.0.24) for Subsection 19.11(i)
Permalink:: http://dlmf.nist.gov/19.11.i
Clarification (effective with 1.0.28):: The unnumbered equation which was inserted in Version 1.0.24 (September 15, 2019) has been given the equation number (19.11.6_5).
Addition (effective with 1.0.24):: A sentence and an unnumbered equation were added at the end of this subsection indicating that care must be taken with the multivalued functions in (19.11.5) (see, e.g., Cayley (1961, pp. 103-106)).
Suggested 2019-01-28 by Albert Groenenboom
See also:: Annotations for §19.11 and Ch.19

19.11.1		$F\left(\theta,k\right)+F\left(\phi,k\right)=F\left(\psi,k\right),$
ⓘ Symbols: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\theta$ : amplitude and $\psi$ : angle Referenced by: §19.11(i) Permalink: http://dlmf.nist.gov/19.11.E1 Encodings: TeX, pMML, png See also: Annotations for §19.11(i), §19.11 and Ch.19

19.11.2		$E\left(\theta,k\right)+E\left(\phi,k\right)=E\left(\psi,k\right)+k^{2}\sin% \theta\sin\phi\sin\psi.$
ⓘ Symbols: $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\theta$ : amplitude and $\psi$ : angle Permalink: http://dlmf.nist.gov/19.11.E2 Encodings: TeX, pMML, png See also: Annotations for §19.11(i), §19.11 and Ch.19

Here

19.11.3	$\displaystyle\sin\psi$	$\displaystyle=\frac{(\sin\theta\cos\phi)\Delta(\phi)+(\sin\phi\cos\theta)% \Delta(\theta)}{1-k^{2}{\sin}^{2}\theta{\sin}^{2}\phi},$
19.11.3	$\displaystyle\Delta(\theta)$	$\displaystyle=\sqrt{1-k^{2}{\sin}^{2}\theta}.$
ⓘ Defines: $\psi$ : angle (locally) and $\Delta(\theta)$ (locally) Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\theta$ : amplitude Referenced by: §19.11(iii) Permalink: http://dlmf.nist.gov/19.11.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.11(i), §19.11 and Ch.19

Also,

19.11.4	$\displaystyle\cos\psi$	$\displaystyle=\frac{\cos\theta\cos\phi-(\sin\theta\sin\phi)\Delta(\theta)% \Delta(\phi)}{1-k^{2}{\sin}^{2}\theta{\sin}^{2}\phi},$
19.11.4	$\displaystyle\tan\left(\tfrac{1}{2}\psi\right)$	$\displaystyle=\frac{(\sin\theta)\Delta(\phi)+(\sin\phi)\Delta(\theta)}{\cos% \theta+\cos\phi}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\theta$ : amplitude, $\psi$ : angle and $\Delta(\theta)$ Permalink: http://dlmf.nist.gov/19.11.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.11(i), §19.11 and Ch.19

Lastly,

19.11.5		$\Pi\left(\theta,\alpha^{2},k\right)+\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left% (\psi,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right),$
ⓘ 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, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude, $\psi$ : angle, $\gamma$ and $\delta$ Referenced by: §19.11(i), §19.11(i), Erratum (V1.0.24) for Subsection 19.11(i) Permalink: http://dlmf.nist.gov/19.11.E5 Encodings: TeX, pMML, png See also: Annotations for §19.11(i), §19.11 and Ch.19

where

19.11.6	$\displaystyle\gamma$	$\displaystyle=(({\csc}^{2}\theta)-\alpha^{2})(({\csc}^{2}\phi)-\alpha^{2})(({% \csc}^{2}\psi)-\alpha^{2}),$
19.11.6	$\displaystyle\delta$	$\displaystyle=\alpha^{2}(1-\alpha^{2})(\alpha^{2}-k^{2}).$
ⓘ Defines: $\gamma$ (locally) and $\delta$ (locally) Symbols: $\csc\NVar{z}$ : cosecant function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude and $\psi$ : angle Permalink: http://dlmf.nist.gov/19.11.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.11(i), §19.11 and Ch.19

In the case of $\theta,\phi\in[0,\pi/2)$ and $0\leq k^{2}\leq\alpha^{2}<\min\left(1,\left(1-\cos\theta\cos\phi\cos\psi\right% )^{-1}\right)$ , we can use

19.11.6_5		$R_{C}\left(\gamma-\delta,\gamma\right)=\frac{-1}{\sqrt{\delta}}\operatorname{% arctan}\left(\frac{\sqrt{\delta}\sin\theta\sin\phi\sin\psi}{\alpha^{2}-1-% \alpha^{2}\cos\theta\cos\phi\cos\psi}\right).$
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos\NVar{z}$ : cosine function, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude, $\psi$ : angle, $\gamma$ and $\delta$ Referenced by: §19.11(i), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5) Permalink: http://dlmf.nist.gov/19.11.E6_5 Encodings: TeX, pMML, png Clarification (effective with 1.0.28): This equation was given a number. See also: Annotations for §19.11(i), §19.11 and Ch.19

Hence, care has to be taken with the multivalued functions in (19.11.5).

§19.11(ii) Case $\psi=\pi/2$

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Referenced by:: §19.36(iv)
Permalink:: http://dlmf.nist.gov/19.11.ii
See also:: Annotations for §19.11 and Ch.19

19.11.7		$F\left(\phi,k\right)=K\left(k\right)-F\left(\theta,k\right),$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : 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, $k$ : real or complex modulus and $\theta$ : amplitude Permalink: http://dlmf.nist.gov/19.11.E7 Encodings: TeX, pMML, png See also: Annotations for §19.11(ii), §19.11 and Ch.19

19.11.8		$E\left(\phi,k\right)=E\left(k\right)-E\left(\theta,k\right)+k^{2}\sin\theta% \sin\phi,$
ⓘ Symbols: $E\left(\NVar{k}\right)$ : 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, $k$ : real or complex modulus and $\theta$ : amplitude Permalink: http://dlmf.nist.gov/19.11.E8 Encodings: TeX, pMML, png See also: Annotations for §19.11(ii), §19.11 and Ch.19

where

19.11.9		$\tan\theta=1/(k^{\prime}\tan\phi).$
ⓘ Symbols: $\tan\NVar{z}$ : tangent function, $\phi$ : real or complex argument, $k^{\prime}$ : complementary modulus and $\theta$ : amplitude Permalink: http://dlmf.nist.gov/19.11.E9 Encodings: TeX, pMML, png See also: Annotations for §19.11(ii), §19.11 and Ch.19

19.11.10		$\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left(\alpha^{2},k\right)-\Pi\left(\theta% ,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right),$
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : 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, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude, $\gamma$ and $\delta$ Permalink: http://dlmf.nist.gov/19.11.E10 Encodings: TeX, pMML, png See also: Annotations for §19.11(ii), §19.11 and Ch.19

where

19.11.11	$\displaystyle\gamma$	$\displaystyle=(1-\alpha^{2})(({\csc}^{2}\theta)-\alpha^{2})(({\csc}^{2}\phi)-% \alpha^{2}),$
19.11.11	$\displaystyle\delta$	$\displaystyle=\alpha^{2}(1-\alpha^{2})(\alpha^{2}-k^{2}).$
ⓘ Defines: $\gamma$ (locally) and $\delta$ (locally) Symbols: $\csc\NVar{z}$ : cosecant function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter and $\theta$ : amplitude Permalink: http://dlmf.nist.gov/19.11.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.11(ii), §19.11 and Ch.19

§19.11(iii) Duplication Formulas

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Keywords:: Legendre’s elliptic integrals, duplication formulas
Permalink:: http://dlmf.nist.gov/19.11.iii
See also:: Annotations for §19.11 and Ch.19

If $\phi=\theta$ in §19.11(i) and $\Delta(\theta)$ is again defined by (19.11.3), then

19.11.12		$F\left(\psi,k\right)=2F\left(\theta,k\right),$
ⓘ Symbols: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $k$ : real or complex modulus, $\theta$ : amplitude and $\psi$ : angle Permalink: http://dlmf.nist.gov/19.11.E12 Encodings: TeX, pMML, png See also: Annotations for §19.11(iii), §19.11 and Ch.19

19.11.13		$E\left(\psi,k\right)=2E\left(\theta,k\right)-k^{2}{\sin}^{2}\theta\sin\psi,$
ⓘ Symbols: $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $k$ : real or complex modulus, $\theta$ : amplitude and $\psi$ : angle Permalink: http://dlmf.nist.gov/19.11.E13 Encodings: TeX, pMML, png See also: Annotations for §19.11(iii), §19.11 and Ch.19

19.11.14		$\sin\psi=(\sin 2\theta)\Delta(\theta)/(1-k^{2}{\sin}^{4}\theta),$
ⓘ Defines: $\psi$ : angle (locally) Symbols: $\sin\NVar{z}$ : sine function, $k$ : real or complex modulus, $\theta$ : amplitude and $\Delta(\theta)$ Permalink: http://dlmf.nist.gov/19.11.E14 Encodings: TeX, pMML, png See also: Annotations for §19.11(iii), §19.11 and Ch.19

19.11.15	$\displaystyle\cos\psi$	$\displaystyle=(\cos\left(2\theta\right)+k^{2}{\sin}^{4}\theta)/(1-k^{2}{\sin}^% {4}\theta),$
	$\displaystyle\tan\left(\tfrac{1}{2}\psi\right)$	$\displaystyle=(\tan\theta)\Delta(\theta),$
	$\displaystyle\sin\theta$	$\displaystyle=(\sin\psi)/\sqrt{(1+\cos\psi)(1+\Delta(\psi))},$
	$\displaystyle\cos\theta$	$\displaystyle=\sqrt{\frac{(\cos\psi)+\Delta(\psi)}{1+\Delta(\psi)}},$
	$\displaystyle\tan\theta$	$\displaystyle=\tan\left(\tfrac{1}{2}\psi\right)\sqrt{\frac{1+\cos\psi}{(\cos% \psi)+\Delta(\psi)}},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $k$ : real or complex modulus, $\theta$ : amplitude, $\Delta(\theta)$ and $\psi$ : angle Permalink: http://dlmf.nist.gov/19.11.E15 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §19.11(iii), §19.11 and Ch.19

19.11.16		$\Pi\left(\psi,\alpha^{2},k\right)=2\Pi\left(\theta,\alpha^{2},k\right)+\alpha^% {2}R_{C}\left(\gamma-\delta,\gamma\right),$
ⓘ 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, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude, $\delta$ , $\psi$ : angle and $\gamma$ Permalink: http://dlmf.nist.gov/19.11.E16 Encodings: TeX, pMML, png See also: Annotations for §19.11(iii), §19.11 and Ch.19

19.11.17	$\displaystyle\gamma$	$\displaystyle=(({\csc}^{2}\theta)-\alpha^{2})^{2}(({\csc}^{2}\psi)-\alpha^{2}),$
19.11.17	$\displaystyle\delta$	$\displaystyle=\alpha^{2}(1-\alpha^{2})(\alpha^{2}-k^{2}).$
ⓘ Defines: $\delta$ (locally) and $\gamma$ (locally) Symbols: $\csc\NVar{z}$ : cosecant function, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\theta$ : amplitude and $\psi$ : angle Permalink: http://dlmf.nist.gov/19.11.E17 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.11(iii), §19.11 and Ch.19

19.10 Relations to Other Functions 19.12 Asymptotic Approximations

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§19.11 Addition Theorems

Contents

§19.11(i) General Formulas

§19.11(ii) Case ψ=π/2

§19.11(iii) Duplication Formulas

§19.11(ii) Case $\psi=\pi/2$