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

§19.11 Addition Theorems

Keywords:: Legendre’s elliptic integrals
Permalink:: http://dlmf.nist.gov/19.11

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

§19.11(i) General Formulas

Notes:: See Byerly (1888, pp. 243–245, 256–258), Edwards (1954, v. 2, pp. 511–513), Cazenave (1969, pp. 83–85). (19.11.5) can be derived from (19.26.9), (19.25.26), and (19.11.1).
Referenced by:: §19.11(iii)
Permalink:: http://dlmf.nist.gov/19.11.i

19.11.1 $\mathop{F\/}\nolimits\!\left(\theta,k\right)+\mathop{F\/}\nolimits\!\left(\phi% ,k\right)=\mathop{F\/}\nolimits\!\left(\psi,k\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument, : real or complex modulus and $\psi$ : angle
Referenced by:: §19.11(i)
Permalink:: http://dlmf.nist.gov/19.11.E1
Encodings:: TeX, pMML, png

19.11.2 $\mathop{E\/}\nolimits\!\left(\theta,k\right)+\mathop{E\/}\nolimits\!\left(\phi% ,k\right)=\mathop{E\/}\nolimits\!\left(\psi,k\right)+k^{2}\mathop{\sin\/}% \nolimits\theta\mathop{\sin\/}\nolimits\phi\mathop{\sin\/}\nolimits\psi.$

Symbols:: $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument, : real or complex modulus and $\psi$ : angle
Permalink:: http://dlmf.nist.gov/19.11.E2
Encodings:: TeX, pMML, png

Here

19.11.3

$\mathop{\sin\/}\nolimits\psi=\frac{(\mathop{\sin\/}\nolimits\theta\mathop{\cos% \/}\nolimits\phi)\Delta(\phi)+(\mathop{\sin\/}\nolimits\phi\mathop{\cos\/}% \nolimits\theta)\Delta(\theta)}{1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta{% \mathop{\sin\/}\nolimits^{{2}}}\phi},$

$\Delta(\theta)=\sqrt{1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument, : real or complex modulus, $\Delta(\theta)$ and $\psi$ : angle
Referenced by:: §19.11(iii)
Permalink:: http://dlmf.nist.gov/19.11.E3
Encodings:: TeX, TeX, pMML, pMML, png, png

Also,

19.11.4

$\mathop{\cos\/}\nolimits\psi=\frac{\mathop{\cos\/}\nolimits\theta\mathop{\cos% \/}\nolimits\phi-(\mathop{\sin\/}\nolimits\theta\mathop{\sin\/}\nolimits\phi)% \Delta(\theta)\Delta(\phi)}{1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta{% \mathop{\sin\/}\nolimits^{{2}}}\phi},$

$\mathop{\tan\/}\nolimits\!\left(\tfrac{1}{2}\psi\right)=\frac{(\mathop{\sin\/}% \nolimits\theta)\Delta(\phi)+(\mathop{\sin\/}\nolimits\phi)\Delta(\theta)}{% \mathop{\cos\/}\nolimits\theta+\mathop{\cos\/}\nolimits\phi}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : tangent function, $\phi$ : real or complex argument, : real or complex modulus, $\Delta(\theta)$ and $\psi$ : angle
Permalink:: http://dlmf.nist.gov/19.11.E4
Encodings:: TeX, TeX, pMML, pMML, png, png

Lastly,

19.11.5 $\mathop{\Pi\/}\nolimits\!\left(\theta,\alpha^{2},k\right)+\mathop{\Pi\/}% \nolimits\!\left(\phi,\alpha^{2},k\right)=\mathop{\Pi\/}\nolimits\!\left(\psi,% \alpha^{2},k\right)-\alpha^{2}\mathop{R_{C}\/}\nolimits\!\left(\gamma-\delta,% \gamma\right),$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\psi$ : angle, $\gamma$ and $\delta$
Referenced by:: §19.11(i)
Permalink:: http://dlmf.nist.gov/19.11.E5
Encodings:: TeX, pMML, png

where

19.11.6

$\gamma=(({\mathop{\csc\/}\nolimits^{{2}}}\theta)-\alpha^{2})(({\mathop{\csc\/}% \nolimits^{{2}}}\phi)-\alpha^{2})(({\mathop{\csc\/}\nolimits^{{2}}}\psi)-% \alpha^{2}),$

$\delta=\alpha^{2}(1-\alpha^{2})(\alpha^{2}-k^{2}).$

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, $\phi$ : real or complex argument, : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\psi$ : angle, $\gamma$ and $\delta$
Permalink:: http://dlmf.nist.gov/19.11.E6
Encodings:: TeX, TeX, pMML, pMML, png, png

§19.11(ii) Case $\psi=\pi/2$

Referenced by:: §19.36(iv)
Permalink:: http://dlmf.nist.gov/19.11.ii

19.11.7 $\mathop{F\/}\nolimits\!\left(\phi,k\right)=\mathop{K\/}\nolimits\!\left(k% \right)-\mathop{F\/}\nolimits\!\left(\theta,k\right),$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument and : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.11.E7
Encodings:: TeX, pMML, png

19.11.8 $\mathop{E\/}\nolimits\!\left(\phi,k\right)=\mathop{E\/}\nolimits\!\left(k% \right)-\mathop{E\/}\nolimits\!\left(\theta,k\right)+k^{2}\mathop{\sin\/}% \nolimits\theta\mathop{\sin\/}\nolimits\phi,$

Symbols:: $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument and : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.11.E8
Encodings:: TeX, pMML, png

where

19.11.9 $\mathop{\tan\/}\nolimits\theta=1/(k^{{\prime}}\mathop{\tan\/}\nolimits\phi).$

Symbols:: $\mathop{\tan\/}\nolimits z$ : tangent function, $\phi$ : real or complex argument and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.11.E9
Encodings:: TeX, pMML, png

19.11.10 $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)=\mathop{\Pi\/}% \nolimits\!\left(\alpha^{2},k\right)-\mathop{\Pi\/}\nolimits\!\left(\theta,% \alpha^{2},k\right)-\alpha^{2}\mathop{R_{C}\/}\nolimits\!\left(\gamma-\delta,% \gamma\right),$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind, $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\gamma$ and $\delta$
Permalink:: http://dlmf.nist.gov/19.11.E10
Encodings:: TeX, pMML, png

where

19.11.11

$\gamma=(1-\alpha^{2})(({\mathop{\csc\/}\nolimits^{{2}}}\theta)-\alpha^{2})(({% \mathop{\csc\/}\nolimits^{{2}}}\phi)-\alpha^{2}),$

$\delta=\alpha^{2}(1-\alpha^{2})(\alpha^{2}-k^{2}).$

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, $\phi$ : real or complex argument, : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\gamma$ and $\delta$
Permalink:: http://dlmf.nist.gov/19.11.E11
Encodings:: TeX, TeX, pMML, pMML, png, png

§19.11(iii) Duplication Formulas

Keywords:: Legendre’s elliptic integrals
Permalink:: http://dlmf.nist.gov/19.11.iii

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

19.11.12 $\mathop{F\/}\nolimits\!\left(\psi,k\right)=2\!\mathop{F\/}\nolimits\!\left(% \theta,k\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, : real or complex modulus and $\psi$ : angle
Permalink:: http://dlmf.nist.gov/19.11.E12
Encodings:: TeX, pMML, png

19.11.13 $\mathop{E\/}\nolimits\!\left(\psi,k\right)=2\!\mathop{E\/}\nolimits\!\left(% \theta,k\right)-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta\mathop{\sin\/}% \nolimits\psi,$

Symbols:: $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\mathop{\sin\/}\nolimits z$ : sine function, : real or complex modulus and $\psi$ : angle
Permalink:: http://dlmf.nist.gov/19.11.E13
Encodings:: TeX, pMML, png

19.11.14 $\mathop{\sin\/}\nolimits\psi=(\mathop{\sin\/}\nolimits 2\theta)\Delta(\theta)/% (1-k^{2}{\mathop{\sin\/}\nolimits^{{4}}}\theta),$

Symbols:: $\mathop{\sin\/}\nolimits z$ : sine function, : real or complex modulus, $\psi$ : angle and $\Delta(\theta)$
Permalink:: http://dlmf.nist.gov/19.11.E14
Encodings:: TeX, pMML, png

19.11.15

$\mathop{\cos\/}\nolimits\psi=(\mathop{\cos\/}\nolimits(2\theta)+k^{2}{\mathop{% \sin\/}\nolimits^{{4}}}\theta)/(1-k^{2}{\mathop{\sin\/}\nolimits^{{4}}}\theta),$

$\mathop{\tan\/}\nolimits\!\left(\tfrac{1}{2}\psi\right)=(\mathop{\tan\/}% \nolimits\theta)\Delta(\theta),$

$\mathop{\sin\/}\nolimits\theta=(\mathop{\sin\/}\nolimits\psi)/\sqrt{(1+\mathop% {\cos\/}\nolimits\psi)(1+\Delta(\psi))},$

$\mathop{\cos\/}\nolimits\theta=\sqrt{\frac{(\mathop{\cos\/}\nolimits\psi)+% \Delta(\psi)}{1+\Delta(\psi)}},$

$\mathop{\tan\/}\nolimits\theta=\mathop{\tan\/}\nolimits\!\left(\tfrac{1}{2}% \psi\right)\sqrt{\frac{1+\mathop{\cos\/}\nolimits\psi}{(\mathop{\cos\/}% \nolimits\psi)+\Delta(\psi)}},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : tangent function, : real or complex modulus, $\psi$ : angle and $\Delta(\theta)$
Permalink:: http://dlmf.nist.gov/19.11.E15
Encodings:: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png

19.11.16 $\mathop{\Pi\/}\nolimits\!\left(\psi,\alpha^{2},k\right)=2\mathop{\Pi\/}% \nolimits\!\left(\theta,\alpha^{2},k\right)+\alpha^{2}\mathop{R_{C}\/}% \nolimits\!\left(\gamma-\delta,\gamma\right),$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : Legendre’s incomplete elliptic integral of the third kind, : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\psi$ : angle, $\gamma$ and $\delta$
Permalink:: http://dlmf.nist.gov/19.11.E16
Encodings:: TeX, pMML, png

19.11.17

$\gamma=(({\mathop{\csc\/}\nolimits^{{2}}}\theta)-\alpha^{2})^{2}(({\mathop{% \csc\/}\nolimits^{{2}}}\psi)-\alpha^{2}),$

$\delta=\alpha^{2}(1-\alpha^{2})(\alpha^{2}-k^{2}).$

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, : real or complex modulus, $\alpha^{2}$ : real or complex parameter, $\psi$ : angle, $\gamma$ and $\delta$
Permalink:: http://dlmf.nist.gov/19.11.E17
Encodings:: TeX, TeX, pMML, pMML, png, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 19.10 Relations to Other Functions 19.12 Asymptotic Approximations

§19.11 Addition Theorems

Contents

§19.11(i) General Formulas

§19.11(ii) Case

§19.11(iii) Duplication Formulas

§19.11(ii) Case $\psi=\pi/2$