19 Elliptic Integrals Legendre’s Integrals 19.5 Maclaurin and Related Expansions 19.7 Connection Formulas

§19.6 Special Cases

Keywords:: Legendre’s elliptic integrals
Permalink:: http://dlmf.nist.gov/19.6

§19.6(i) Complete Elliptic Integrals
§19.6(ii) $\mathop{F\/}\nolimits\!\left(\phi,k\right)$
§19.6(iii) $\mathop{E\/}\nolimits\!\left(\phi,k\right)$
§19.6(iv) $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$
§19.6(v) $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$

§19.6(i) Complete Elliptic Integrals

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
Permalink:: http://dlmf.nist.gov/19.6.i

19.6.1

$\mathop{K\/}\nolimits\!\left(0\right)=\mathop{E\/}\nolimits\!\left(0\right)=% \mathop{{K^{{\prime}}}\/}\nolimits\!\left(1\right)=\mathop{{E^{{\prime}}}\/}% \nolimits\!\left(1\right)=\tfrac{1}{2}\pi,$

$\mathop{K\/}\nolimits\!\left(1\right)=\mathop{{K^{{\prime}}}\/}\nolimits\!% \left(0\right)=\infty,$

$\mathop{E\/}\nolimits\!\left(1\right)=\mathop{{E^{{\prime}}}\/}\nolimits\!% \left(0\right)=1.$

Symbols:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{{E^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the second kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind and $\mathop{E\/}\nolimits\!\left(k\right)$ : 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

19.6.2

$\mathop{\Pi\/}\nolimits\!\left(k^{2},k\right)=\mathop{E\/}\nolimits\!\left(k% \right)/{k^{{\prime}}}^{2},$ $k^{2}<1$ ,

$\mathop{\Pi\/}\nolimits\!\left(-k,k\right)=\tfrac{1}{4}\pi(1+k)^{{-1}}+\tfrac{% 1}{2}\mathop{K\/}\nolimits\!\left(k\right),$ $0\leq k^{2}<1$ .

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind, : 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

19.6.3 $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},0\right)=\pi/(2\sqrt{1-\alpha^{2}}),% \quad\mathop{\Pi\/}\nolimits\!\left(0,k\right)=\mathop{K\/}\nolimits\!\left(k% \right),$ $-\infty<\alpha^{2}<1$ .

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind, : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/19.6.E3
Encodings:: TeX, pMML, png

19.6.4

$\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)\to+\infty,$ $\alpha^{2}\to 1-$ ,

$\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)\to\infty\mathop{\mathrm{% sign}\/}\nolimits\!\left(1-\alpha^{2}\right),$ $k^{2}\to 1-$ .

Symbols:: $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind, $\mathop{\mathrm{sign}\/}\nolimits x$ : sign of , : 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

If $1<\alpha^{2}<\infty$ , then the Cauchy principal value satisfies

19.6.5 $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)=\mathop{K\/}\nolimits\!% \left(k\right)-\mathop{\Pi\/}\nolimits\!\left(k^{2}/\alpha^{2},k\right),$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind, : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.12, 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

and

19.6.6

$\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},0\right)=0,$

$\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)\to\mathop{K\/}\nolimits\!% \left(k\right)-\left(\mathop{E\/}\nolimits\!\left(k\right)/{k^{{\prime}}}^{2}% \right),$ $\alpha^{2}\to 1+$ ,

$\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)\to-\infty,$ $k^{2}\rightarrow 1-$ .

Exact values of $\mathop{K\/}\nolimits\!\left(k\right)$ and $\mathop{E\/}\nolimits\!\left(k\right)$ for various special values of are given in Byrd and Friedman (1971, 111.10 and 111.11) and Cooper et al. (2006).

§19.6(ii) $\mathop{F\/}\nolimits\!\left(\phi,k\right)$

Notes:: Use (19.2.4), (19.16.6), and (19.25.5).
Keywords:: Legendre’s elliptic integrals, $\mathop{R_{C}\/}\nolimits$ -function, inverse Gudermannian function
Permalink:: http://dlmf.nist.gov/19.6.ii

19.6.7

$\mathop{F\/}\nolimits\!\left(0,k\right)=0,$

$\mathop{F\/}\nolimits\!\left(\phi,0\right)=\phi,$

$\mathop{F\/}\nolimits\!\left(\tfrac{1}{2}\pi,1\right)=\infty,$

$\mathop{F\/}\nolimits\!\left(\tfrac{1}{2}\pi,k\right)=\mathop{K\/}\nolimits\!% \left(k\right),$

$\lim_{{\phi\to 0}}\ifrac{\mathop{F\/}\nolimits\!\left(\phi,k\right)}{\phi}=1.$

19.6.8 $\mathop{F\/}\nolimits\!\left(\phi,1\right)=(\mathop{\sin\/}\nolimits\phi)% \mathop{R_{C}\/}\nolimits\!\left(1,{\mathop{\cos\/}\nolimits^{{2}}}\phi\right)% =\mathop{{\mathrm{gd}^{{-1}}}\/}\nolimits\!\left(\phi\right).$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{{\mathrm{gd}^{{-1}}}\/}\nolimits\!\left(x\right)$ : inverse Gudermannian function, $\mathop{\sin\/}\nolimits z$ : 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

For the inverse Gudermannian function $\mathop{{\mathrm{gd}^{{-1}}}\/}\nolimits\!\left(\phi\right)$ see §4.23(viii). Compare also (19.10.2).

§19.6(iii) $\mathop{E\/}\nolimits\!\left(\phi,k\right)$

Notes:: Use (19.2.5).
Permalink:: http://dlmf.nist.gov/19.6.iii

19.6.9

$\mathop{E\/}\nolimits\!\left(0,k\right)=0,$

$\mathop{E\/}\nolimits\!\left(\phi,0\right)=\phi,$

$\mathop{E\/}\nolimits\!\left(\tfrac{1}{2}\pi,1\right)=1,$

$\mathop{E\/}\nolimits\!\left(\phi,1\right)=\mathop{\sin\/}\nolimits\phi,$

$\mathop{E\/}\nolimits\!\left(\tfrac{1}{2}\pi,k\right)=\mathop{E\/}\nolimits\!% \left(k\right).$

19.6.10 $\lim_{{\phi\to 0}}\ifrac{\mathop{E\/}\nolimits\!\left(\phi,k\right)}{\phi}=1.$

Symbols:: $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\phi$ : real or complex argument and : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.6.E10
Encodings:: TeX, pMML, png

§19.6(iv) $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$

Notes:: Byrd and Friedman (1971, 111.01 and 111.04, p. 10) also needs $\alpha\mathop{\sin\/}\nolimits\phi<1$ . 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

Circular and hyperbolic cases, including Cauchy principal values, are unified by using $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ . Let $c={\mathop{\csc\/}\nolimits^{{2}}}\phi\neq\alpha^{2}$ and $\Delta=\sqrt{1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\phi}$ . Then

19.6.11

$\mathop{\Pi\/}\nolimits\!\left(0,\alpha^{2},k\right)=0,$

$\mathop{\Pi\/}\nolimits\!\left(\phi,0,0\right)=\phi,$

$\mathop{\Pi\/}\nolimits\!\left(\phi,1,0\right)=\mathop{\tan\/}\nolimits\phi.$

Symbols:: $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\mathop{\tan\/}\nolimits z$ : tangent function, $\phi$ : real or complex argument, : 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

19.6.12

$\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},0\right)=\mathop{R_{C}\/}% \nolimits\!\left(c-1,c-\alpha^{2}\right),$

$\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},1\right)=\frac{1}{1-\alpha^{2}}% \left(\mathop{R_{C}\/}\nolimits\!\left(c,c-1\right)-\alpha^{2}\mathop{R_{C}\/}% \nolimits\!\left(c,c-\alpha^{2}\right)\right),$

$\mathop{\Pi\/}\nolimits\!\left(\phi,1,1\right)=\tfrac{1}{2}(\mathop{R_{C}\/}% \nolimits\!\left(c,c-1\right)+\sqrt{c}(c-1)^{{-1}}).$

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 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

19.6.13

$\mathop{\Pi\/}\nolimits\!\left(\phi,0,k\right)=\mathop{F\/}\nolimits\!\left(% \phi,k\right),$

$\mathop{\Pi\/}\nolimits\!\left(\phi,k^{2},k\right)=\frac{1}{{k^{{\prime}}}^{2}% }\left(\mathop{E\/}\nolimits\!\left(\phi,k\right)-\frac{k^{2}}{\Delta}\mathop{% \sin\/}\nolimits\phi\mathop{\cos\/}\nolimits\phi\right),$

$\mathop{\Pi\/}\nolimits\!\left(\phi,1,k\right)=\mathop{F\/}\nolimits\!\left(% \phi,k\right)-\frac{1}{{k^{{\prime}}}^{2}}(\mathop{E\/}\nolimits\!\left(\phi,k% \right)-\Delta\mathop{\tan\/}\nolimits\phi).$

19.6.14

$\mathop{\Pi\/}\nolimits\!\left(\tfrac{1}{2}\pi,\alpha^{2},k\right)=\mathop{\Pi% \/}\nolimits\!\left(\alpha^{2},k\right),$

$\lim_{{\phi\to 0}}\ifrac{\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k% \right)}{\phi}=1.$

Symbols:: $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : 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 and $\alpha^{2}$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/19.6.E14
Encodings:: TeX, TeX, pMML, pMML, png, png

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

§19.6(v) $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$

Keywords:: $\mathop{R_{C}\/}\nolimits$ -function
Permalink:: http://dlmf.nist.gov/19.6.v

19.6.15

$\mathop{R_{C}\/}\nolimits\!\left(x,x\right)=x^{{-1/2}},$

$\mathop{R_{C}\/}\nolimits\!\left(\lambda x,\lambda y\right)=\lambda^{{-1/2}}% \mathop{R_{C}\/}\nolimits\!\left(x,y\right),$

$\mathop{R_{C}\/}\nolimits\!\left(x,y\right)\to+\infty,$ $y\to 0+$ or $y\to 0-$ ,

$\mathop{R_{C}\/}\nolimits\!\left(0,y\right)=\tfrac{1}{2}\pi y^{{-1/2}},$ $|\mathop{\mathrm{ph}\/}\nolimits y|<\pi$ ,

$\mathop{R_{C}\/}\nolimits\!\left(0,y\right)=0,$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\mathop{\mathrm{ph}\/}\nolimits$ : phase
Referenced by:: §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

© 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. 19.5 Maclaurin and Related Expansions 19.7 Connection Formulas

§19.6 Special Cases

Contents

§19.6(i) Complete Elliptic Integrals

§19.6(ii)

§19.6(iii)

§19.6(iv)

§19.6(v)

§19.6(ii) $\mathop{F\/}\nolimits\!\left(\phi,k\right)$

§19.6(iii) $\mathop{E\/}\nolimits\!\left(\phi,k\right)$

§19.6(iv) $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$

§19.6(v) $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$