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11: 19.4 Derivatives and Differential Equations

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19.4.5 $$\frac{\partial F(\varphi ,k)}{\partial k}=\frac{E(\varphi ,k)-k_{}^{\prime }{}_{}{}^{2}F(\varphi ,k)}{kk_{}^{\prime }{}_{}{}^2}-\frac{k\sin \varphi \cos \varphi }{k_{}^{\prime }{}_{}{}^2\sqrt{1-k^2{\sin }^2\varphi }},$$

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.7 $$\frac{\partial \mathrm{\Pi }(\varphi ,{\alpha }^2,k)}{\partial k}=\frac{k}{k_{}^{\prime }{}_{}{}^2(k^2-{\alpha }^2)}\left(E(\varphi ,k)-k_{}^{\prime }{}_{}{}^2\mathrm{\Pi }(\varphi ,{\alpha }^2,k)-\frac{k^2\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}\right).$$

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Symbols:

$\cos z$: cosine function, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Permalink:

http://dlmf.nist.gov/19.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.8 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+(1-3k^2)D_k-k)F(\varphi ,k)=\frac{-k\sin \varphi \cos \varphi }{{(1-k^2{\sin }^2\varphi )}^{3/2}},$$

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Referenced by:

§19.4(ii)

Permalink:

http://dlmf.nist.gov/19.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

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19.4.9 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+k_{}^{\prime }{}_{}{}^2{D}_k+k)E(\varphi ,k)=\frac{k\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}.$$

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Symbols:

$\cos z$: cosine function, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Permalink:

http://dlmf.nist.gov/19.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

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12: 33.11 Asymptotic Expansions for Large $\rho$

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$F_{\mathrm{\ell }}(\eta ,\rho )=g(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}+f(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$

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$G_{\mathrm{\ell }}(\eta ,\rho )=f(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}-g(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$

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$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )=\widehat{g}(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}+\widehat{f}(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$

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$G_{\mathrm{\ell }}^{\prime }(\eta ,\rho )=\widehat{f}(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}-\widehat{g}(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$

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33.11.7 $$g(\eta ,\rho )\widehat{f}(\eta ,\rho )-f(\eta ,\rho )\widehat{g}(\eta ,\rho )=1.$$

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Symbols:

$\rho$: nonnegative real variable, $\eta$: real parameter, $f(\eta ,\rho )$: function, $g(\eta ,\rho )$: function, $\widehat{f}(\eta ,\rho )$: function and $\widehat{g}(\eta ,\rho )$: function

A&S Ref:

14.5.8

Referenced by:

§33.11, Erratum (V1.0.22) for Equations (33.11.2)–(33.11.7), Erratum (V1.0.22) for Equations (33.11.2)–(33.11.7)

Permalink:

http://dlmf.nist.gov/33.11.E7

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.22):

The arguments of some of the functions in this equation were included to improve clarity of the presentation.

See also:

Annotations for §33.11 and Ch.33

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13: 19.30 Lengths of Plane Curves

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19.30.2 $$s=a{\int }_0^{\varphi }\sqrt{1-k^2{\sin }^2\theta }d\theta .$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $s$: arclength

Permalink:

http://dlmf.nist.gov/19.30.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.30(i), §19.30 and Ch.19

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19.30.6 $$\frac{\partial s}{\partial (1/k)}=\sqrt{a^2-b^2}F(\varphi ,k)=\sqrt{a^2-b^2}{R}_F(c-1,c-k^2,c),$$ $k^2=(a^2-b^2)/(a^2+\lambda )$, $c={\csc }^2\varphi$.

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\csc z$: cosecant function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\varphi$: real or complex argument, $k$: real or complex modulus and $s$: arclength

Referenced by:

§19.30(i)

Permalink:

http://dlmf.nist.gov/19.30.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.30(i), §19.30 and Ch.19

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19.30.12 $$s=aF(\varphi ,1/\sqrt{2}),$$ $\varphi =\arcsin \sqrt{2/(q+1)}=\arccos \left(\tan \theta \right)$.

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\arccos z$: arccosine function, $\arcsin z$: arcsine function, $\tan z$: tangent function, $\varphi$: real or complex argument, $q$ and $s$: arclength

Permalink:

http://dlmf.nist.gov/19.30.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.30(iii), §19.30 and Ch.19

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14: 19.9 Inequalities

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19.9.12 $$\max (\sin \varphi ,\varphi \mathrm{\Delta })\le E(\varphi ,k)\le \varphi ,$$

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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 and $\mathrm{\Delta }$

Permalink:

http://dlmf.nist.gov/19.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

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19.9.14

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\mathrm{\Delta }$

Referenced by:

§19.9(ii), §19.9(ii)

Permalink:

http://dlmf.nist.gov/19.9.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

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19.9.15

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\mathrm{\Delta }$

Referenced by:

§19.9(ii), §19.9(ii)

Permalink:

http://dlmf.nist.gov/19.9.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

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19.9.16 $$F(\varphi ,k)=\frac{2}{\pi }K\left(k^{\prime }\right)\ln \left(\frac{4}{\mathrm{\Delta }+\cos \varphi }\right)-\theta {\mathrm{\Delta }}^2,$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $\mathrm{\Delta }$

Referenced by:

§19.9(ii)

Permalink:

http://dlmf.nist.gov/19.9.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

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15: 19.8 Quadratic Transformations

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$${\varphi }_1=\varphi +\arctan \left(k^{\prime }\tan \varphi \right)=\arcsin \left((1+k^{\prime })\frac{\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}\right).$$

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$$2{\varphi }_2=\varphi +\arcsin \left(k\sin \varphi \right).$$

… ►We consider only the descending Gauss transformation because its (ascending) inverse moves $F(\varphi ,k)$ closer to the singularity at $k=\sin \varphi =1$. … ►

$\sin {\psi }_1={\displaystyle \frac{(1+k^{\prime })\sin \varphi }{1+\mathrm{\Delta }}},$

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$c={\csc }^2\varphi .$

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16: 19.33 Triaxial Ellipsoids

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19.33.2 $$\frac{S}{2\pi }=c^2+\frac{ab}{\sin \varphi }\left(E(\varphi ,k){\sin }^2\varphi +F(\varphi ,k){\cos }^2\varphi \right),$$ $a\ge b\ge c$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first 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 $S$: area

Permalink:

http://dlmf.nist.gov/19.33.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(i), §19.33 and Ch.19

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17: 19.6 Special Cases

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19.6.8 $$F(\varphi ,1)=(\sin \varphi )R_C(1,{\cos }^2\varphi )={\mathrm{gd}}^{-1}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.10(ii)

Permalink:

http://dlmf.nist.gov/19.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.6(ii), §19.6 and Ch.19

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18: 19.10 Relations to Other Functions

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19.10.2 $$(\sinh \varphi )R_C(1,{\cosh }^2\varphi )=\mathrm{gd}\left(\varphi \right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{gd}x$: Gudermannian function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $\varphi$: real or complex argument

Referenced by:

§19.10(ii), §19.6(ii)

Permalink:

http://dlmf.nist.gov/19.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.10(ii), §19.10 and Ch.19

19: 19.3 Graphics

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Figure 19.3.3: $F(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. … Magnify 3D Help

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Figure 19.3.4: $E(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. … Magnify 3D Help

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Figure 19.3.6: $\mathrm{\Pi }(\varphi ,2,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 3$, . Cauchy principal values are shown when ${\sin }^2\varphi >\frac{1}{2}$. … Magnify 3D Help

… ►In Figures 19.3.7 and 19.3.8 for complete Legendre’s elliptic integrals with complex arguments, height corresponds to the absolute value of the function and color to the phase. …

20: 4.46 Tables

§4.46 Tables

►Extensive numerical tables of all the elementary functions for real values of their arguments appear in Abramowitz and Stegun (1964, Chapter 4). … ►For 40D values of the first 500 roots of $\tan x=x$, see Robinson (1972). … ►For 10S values of the first five complex roots of $\sin z=az$, $\cos z=az$, and $\cosh z=az$, for selected positive values of $a$, see Fettis (1976). …