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real argument

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11: 12.21 Software

§12.21(ii) Real Arguments and Parameters

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12: 13.32 Software

§13.32(ii) Real Argument and Parameters

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13: 5.3 Graphics

§5.3(i) Real Argument

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14: Software Index

… в–є в–є в–є в–є в–є в–є в–є в–є

	Open Source													With Book						Commercial
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16.27(ii) Real Arguments	вњ“		вњ“		вњ“	вњ“						a	вњ“								вњ“	вњ“	вњ“
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20.16(ii) Real arguments	вњ“											a	вњ“	вњ“			вњ“				вњ“	вњ“	a	вњ“
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22.22(ii) Real Argument	вњ“		вњ“		вњ“	вњ“		вњ“				вњ“	вњ“	вњ“		вњ“	вњ“		вњ“	вњ“	вњ“	вњ“	вњ“	вњ“
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23.24(ii) Real Argument																					вњ“	вњ“	a
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33.26(ii) Real arguments	вњ“				вњ“				вњ“			a	вњ“	вњ“			вњ“				вњ“		a
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15: 25.21 Software

§25.21(ii) Zeta Functions for Real Arguments

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16: 4.1 Special Notation

… в–єIt is assumed the user is familiar with the definitions and properties of elementary functions of real arguments

x

. …

17: 19.11 Addition Theorems

19.11.1

F\left(\theta,k\right)+F\left(\phi,k\right)=F\left(\psi,k\right),

в“

Symbols:: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : 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:: pMML, png, TeX
See also:: Annotations for §19.11(i), §19.11 and Ch.19

в–є

19.11.2

E\left(\theta,k\right)+E\left(\phi,k\right)=E\left(\psi,k\right)+k^{2}\sin% \theta\sin\phi\sin\psi.

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Symbols:: $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\theta$ : amplitude and $\psi$ : angle
Permalink:: http://dlmf.nist.gov/19.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(i), §19.11 and Ch.19

… в–є

19.11.7

F\left(\phi,k\right)=K\left(k\right)-F\left(\theta,k\right),

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Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\theta$ : amplitude
Permalink:: http://dlmf.nist.gov/19.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(ii), §19.11 and Ch.19

в–є

19.11.8

E\left(\phi,k\right)=E\left(k\right)-E\left(\theta,k\right)+k^{2}\sin\theta% \sin\phi,

в“

Symbols:: $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\theta$ : amplitude
Permalink:: http://dlmf.nist.gov/19.11.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(ii), §19.11 and Ch.19

… в–є

19.11.9

\tan\theta=1/(k^{\prime}\tan\phi).

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Symbols:: $\tan\NVar{z}$ : tangent function, $\phi$ : real or complex argument, $k^{\prime}$ : complementary modulus and $\theta$ : amplitude
Permalink:: http://dlmf.nist.gov/19.11.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.11(ii), §19.11 and Ch.19

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18: 14.34 Software

§14.34(ii) Legendre Functions: Real Argument and Parameters

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19: 19.4 Derivatives and Differential Equations

19.4.5

\frac{\partial F\left(\phi,k\right)}{\partial k}={\frac{E\left(\phi,k\right)-{% k^{\prime}}^{2}F\left(\phi,k\right)}{k{k^{\prime}}^{2}}-\frac{k\sin\phi\cos% \phi}{{k^{\prime}}^{2}\sqrt{1-k^{2}{\sin}^{2}\phi}}},

в“

Symbols:: $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\sin\NVar{z}$ : sine function, $\phi$ : 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

в–є

19.4.6

\frac{\partial E\left(\phi,k\right)}{\partial k}=\frac{E\left(\phi,k\right)-F% \left(\phi,k\right)}{k},

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Symbols:: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\phi$ : real or complex argument and $k$ : real or complex modulus
Referenced by:: §19.30(i)
Permalink:: http://dlmf.nist.gov/19.4.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.4(i), §19.4 and Ch.19

в–є

19.4.7

\frac{\partial\Pi\left(\phi,\alpha^{2},k\right)}{\partial k}=\frac{k}{{k^{% \prime}}^{2}(k^{2}-\alpha^{2})}\left({E\left(\phi,k\right)-{k^{\prime}}^{2}\Pi% \left(\phi,\alpha^{2},k\right)}-\frac{k^{2}\sin\phi\cos\phi}{\sqrt{1-k^{2}{% \sin}^{2}\phi}}\right).

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Symbols:: $\cos\NVar{z}$ : cosine function, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\sin\NVar{z}$ : sine function, $\phi$ : 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

… в–є

19.4.8

(k{k^{\prime}}^{2}D_{k}^{2}+(1-3k^{2})D_{k}-k)F\left(\phi,k\right)=\frac{-k% \sin\phi\cos\phi}{(1-k^{2}{\sin}^{2}\phi)^{3/2}},

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Symbols:: $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\sin\NVar{z}$ : sine function, $\phi$ : 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

в–є

19.4.9

(k{k^{\prime}}^{2}D_{k}^{2}+{k^{\prime}}^{2}D_{k}+k)E\left(\phi,k\right)=\frac% {k\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin}^{2}\phi}}.

в“

Symbols:: $\cos\NVar{z}$ : cosine function, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : 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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20: 9.19 Approximations

… в–єThese expansions are for real arguments

x

and are supplied in sets of four for each function, corresponding to intervals

-\infty<x\leq a

,

a\leq x\leq 0

,

0\leq x\leq b

,

b\leq x<\infty

. …

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