§19.11 Addition Theorems

19.11.1		$$F(\theta ,k)+F(\varphi ,k)=F(\psi ,k),$$
ⓘ Symbols: $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: 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		$$E(\theta ,k)+E(\varphi ,k)=E(\psi ,k)+k^2\sin \theta \sin \varphi \sin \psi .$$
ⓘ Symbols: $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: 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		$\sin \psi$	$={\displaystyle \frac{(\sin \theta \cos \varphi )\mathrm{\Delta }(\varphi )+(\sin \varphi \cos \theta )\mathrm{\Delta }(\theta )}{1-k^2{\sin }^2\theta {\sin }^2\varphi }},$
		$\mathrm{\Delta }(\theta )$	$=\sqrt{1-k^2{\sin }^2\theta }.$
ⓘ Defines: $\psi$: angle (locally) and $\mathrm{\Delta }(\theta )$ (locally) Symbols: $\cos z$: cosine function, $\sin z$: sine function, $\varphi$: 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		$\cos \psi$	$={\displaystyle \frac{\cos \theta \cos \varphi -(\sin \theta \sin \varphi )\mathrm{\Delta }(\theta )\mathrm{\Delta }(\varphi )}{1-k^2{\sin }^2\theta {\sin }^2\varphi }},$
		$\tan \left(\frac{1}{2}\psi \right)$	$={\displaystyle \frac{(\sin \theta )\mathrm{\Delta }(\varphi )+(\sin \varphi )\mathrm{\Delta }(\theta )}{\cos \theta +\cos \varphi }}.$
ⓘ Symbols: $\cos z$: cosine function, $\sin z$: sine function, $\tan z$: tangent function, $\varphi$: real or complex argument, $k$: real or complex modulus, $\theta$: amplitude, $\psi$: angle and $\mathrm{\Delta }(\theta )$ 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		$$\mathrm{\Pi }(\theta ,{\alpha }^2,k)+\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\mathrm{\Pi }(\psi ,{\alpha }^2,k)-{\alpha }^2{R}_C(\gamma -\delta ,\gamma ),$$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: 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		$\gamma$	$=(({\csc }^2\theta )-{\alpha }^2)(({\csc }^2\varphi )-{\alpha }^2)(({\csc }^2\psi )-{\alpha }^2),$
		$\delta$	$={\alpha }^2(1-{\alpha }^2)({\alpha }^2-k^2).$
ⓘ Defines: $\gamma$ (locally) and $\delta$ (locally) Symbols: $\csc z$: cosecant function, $\varphi$: 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 $\theta ,\varphi \in [0,\pi /2)$ and , we can use

19.11.6_5		$$R_C(\gamma -\delta ,\gamma )=\frac{-1}{\sqrt{\delta }}\arctan \left(\frac{\sqrt{\delta }\sin \theta \sin \varphi \sin \psi }{{\alpha }^2-1-{\alpha }^2\cos \theta \cos \varphi \cos \psi }\right).$$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $\arctan z$: arctangent function, $\sin z$: sine function, $\varphi$: 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).

19.11.7		$$F(\varphi ,k)=K\left(k\right)-F(\theta ,k),$$
ⓘ Symbols: $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: 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		$$E(\varphi ,k)=E\left(k\right)-E(\theta ,k)+k^2\sin \theta \sin \varphi ,$$
ⓘ Symbols: $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: 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		$$\tan \theta =1/(k^{\prime }\tan \varphi ).$$
ⓘ Symbols: $\tan z$: tangent function, $\varphi$: 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		$$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\mathrm{\Pi }({\alpha }^2,k)-\mathrm{\Pi }(\theta ,{\alpha }^2,k)-{\alpha }^2{R}_C(\gamma -\delta ,\gamma ),$$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: 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		$\gamma$	$=(1-{\alpha }^2)(({\csc }^2\theta )-{\alpha }^2)(({\csc }^2\varphi )-{\alpha }^2),$
		$\delta$	$={\alpha }^2(1-{\alpha }^2)({\alpha }^2-k^2).$
ⓘ Defines: $\gamma$ (locally) and $\delta$ (locally) Symbols: $\csc z$: cosecant function, $\varphi$: 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

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

19.11.12		$$F(\psi ,k)=2F(\theta ,k),$$
ⓘ Symbols: $F(\varphi ,k)$: 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		$$E(\psi ,k)=2E(\theta ,k)-k^2{\sin }^2\theta \sin \psi ,$$
ⓘ Symbols: $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: 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		$$\sin \psi =(\sin 2\theta )\mathrm{\Delta }(\theta )/(1-k^2{\sin }^4\theta ),$$
ⓘ Defines: $\psi$: angle (locally) Symbols: $\sin z$: sine function, $k$: real or complex modulus, $\theta$: amplitude and $\mathrm{\Delta }(\theta )$ 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		$\cos \psi$	$=(\cos \left(2\theta \right)+k^2{\sin }^4\theta )/(1-k^2{\sin }^4\theta ),$
		$\tan \left(\frac{1}{2}\psi \right)$	$=(\tan \theta )\mathrm{\Delta }(\theta ),$
		$\sin \theta$	$=(\sin \psi )/\sqrt{(1+\cos \psi )(1+\mathrm{\Delta }(\psi ))},$
		$\cos \theta$	$=\sqrt{{\displaystyle \frac{(\cos \psi )+\mathrm{\Delta }(\psi )}{1+\mathrm{\Delta }(\psi )}}},$
		$\tan \theta$	$=\tan \left(\frac{1}{2}\psi \right)\sqrt{{\displaystyle \frac{1+\cos \psi }{(\cos \psi )+\mathrm{\Delta }(\psi )}}},$
ⓘ Symbols: $\cos z$: cosine function, $\sin z$: sine function, $\tan z$: tangent function, $k$: real or complex modulus, $\theta$: amplitude, $\mathrm{\Delta }(\theta )$ 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		$$\mathrm{\Pi }(\psi ,{\alpha }^2,k)=2\mathrm{\Pi }(\theta ,{\alpha }^2,k)+{\alpha }^2{R}_C(\gamma -\delta ,\gamma ),$$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: 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		$\gamma$	$={(({\csc }^2\theta )-{\alpha }^2)}^2(({\csc }^2\psi )-{\alpha }^2),$
		$\delta$	$={\alpha }^2(1-{\alpha }^2)({\alpha }^2-k^2).$
ⓘ Defines: $\delta$ (locally) and $\gamma$ (locally) Symbols: $\csc z$: 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