§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
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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, $k$ : real or complex modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §19.6(i)
Permalink:: http://dlmf.nist.gov/19.6.E2
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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, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/19.6.E3
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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 $x$ , $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/19.6.E4
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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, $k$ : 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
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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 $k$ 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
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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 $k$ : real or complex modulus
Permalink:: http://dlmf.nist.gov/19.6.E10
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§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, $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/19.6.E11
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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
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19.6.13

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

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

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