real argument

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21: 19.9 Inequalities

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19.9.11 $$\varphi \le F(\varphi ,k)\le \min (\varphi /\mathrm{\Delta },{\mathrm{gd}}^{-1}\left(\varphi \right)),$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\mathrm{\Delta }$

Permalink:

http://dlmf.nist.gov/19.9.E11

Encodings:

pMML, png, TeX

See also:

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

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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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19.9.13 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,0)\le \mathrm{\Pi }(\varphi ,{\alpha }^2,k)\le \min (\mathrm{\Pi }(\varphi ,{\alpha }^2,0)/\mathrm{\Delta },\mathrm{\Pi }(\varphi ,{\alpha }^2,1)).$$

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Symbols:

$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter and $\mathrm{\Delta }$

Referenced by:

§19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E13

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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19.9.17 $$L\le F(\varphi ,k)\le \sqrt{UL}\le \frac{1}{2}(U+L)\le U,$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $L$: lower limit and $U$: upper limit

Permalink:

http://dlmf.nist.gov/19.9.E17

Encodings:

pMML, png, TeX

See also:

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

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22: 19.14 Reduction of General Elliptic Integrals

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19.14.5 $${\sin }^2\varphi =\frac{\gamma -\alpha }{U^2+\gamma },$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.7 $${\sin }^2\varphi =\frac{(\gamma -\alpha )x^2}{a_1{a}_2+\gamma x^2}.$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.8 $${\sin }^2\varphi =\frac{\gamma -\alpha }{b_1{b}_2{y}^2+\gamma }.$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.9 $${\sin }^2\varphi =\frac{(\gamma -\alpha )(x^2-y^2)}{\gamma (x^2-y^2)-a_1(a_2+b_2{x}^2)}.$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Permalink:

http://dlmf.nist.gov/19.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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19.14.10 $${\sin }^2\varphi =\frac{(\gamma -\alpha )(y^2-x^2)}{\gamma (y^2-x^2)-a_1(a_2+b_2{y}^2)}.$$

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Symbols:

$\sin z$: sine function and $\varphi$: real or complex argument

Referenced by:

§19.15, §19.29(iii)

Permalink:

http://dlmf.nist.gov/19.14.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.14(i), §19.14 and Ch.19

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23: 19.7 Connection Formulas

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19.7.8 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)+\mathrm{\Pi }(\varphi ,{\omega }^2,k)=F(\varphi ,k)+\sqrt{c}{R}_C((c-1)(c-k^2),(c-{\alpha }^2)(c-{\omega }^2)),$$ ${\alpha }^2{\omega }^2=k^2$.

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Defines:

${\omega }^2$: change of variable (locally)

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter and $c$: change of variable

Referenced by:

§19.25(i), §19.6(i)

Permalink:

http://dlmf.nist.gov/19.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.7(iii), §19.7 and Ch.19

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19.7.9 $$(k^2-{\alpha }^2)\mathrm{\Pi }(\varphi ,{\alpha }^2,k)+(k^2-{\omega }^2)\mathrm{\Pi }(\varphi ,{\omega }^2,k)=k^{2}F(\varphi ,k)-{\alpha }^2{\omega }^2\sqrt{c-1}{R}_C(c(c-k^2),(c-{\alpha }^2)(c-{\omega }^2)),$$ $(1-{\alpha }^2)(1-{\omega }^2)=1-k^2$.

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Defines:

${\omega }^2$: change of variable (locally)

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter and $c$: change of variable

Permalink:

http://dlmf.nist.gov/19.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.7(iii), §19.7 and Ch.19

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19.7.10 $$(1-{\alpha }^2)\mathrm{\Pi }(\varphi ,{\alpha }^2,k)+(1-{\omega }^2)\mathrm{\Pi }(\varphi ,{\omega }^2,k)=F(\varphi ,k)+(1-{\alpha }^2-{\omega }^2)\sqrt{c-k^2}{R}_C(c(c-1),(c-{\alpha }^2)(c-{\omega }^2)),$$ $(k^2-{\alpha }^2)(k^2-{\omega }^2)=k^2(k^2-1)$.

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Defines:

${\omega }^2$: change of variable (locally)

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter and $c$: change of variable

Permalink:

http://dlmf.nist.gov/19.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.7(iii), §19.7 and Ch.19

24: 19.8 Quadratic Transformations

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19.8.14 $$2(k^2-{\alpha }^2)\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\frac{{\omega }^2-{\alpha }^2}{1+k^{\prime }}\mathrm{\Pi }({\varphi }_1,{\alpha }_1^2,k_1)+k^{2}F(\varphi ,k)-(1+k^{\prime }){\alpha }_1^2{R}_C(c_1,c_1-{\alpha }_1^2),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter, $k_1$: change of variable, ${\varphi }_1$: change of variable, $\omega$: change of variable, ${\alpha }_1$: change of variable and $c_1$: change of variable

Permalink:

http://dlmf.nist.gov/19.8.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(ii), §19.8(ii), §19.8 and Ch.19

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19.8.20 $$\rho \mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\frac{4}{1+k^{\prime }}\mathrm{\Pi }({\psi }_1,{\alpha }_1^2,k_1)+(\rho -1)F(\varphi ,k)-R_C(c-1,c-{\alpha }^2),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter, $k_1$: change of variable, ${\psi }_1$: change of variable, $\rho$: change of variable, ${\alpha }_1^2$: change of variable and $c$: change of variable

Referenced by:

§19.36(ii)

Permalink:

http://dlmf.nist.gov/19.8.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(iii), §19.8 and Ch.19

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25: 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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19.30.3 $$s/a=E(\varphi ,k)=R_F(c-1,c-k^2,c)-\frac{1}{3}{k}^2{R}_D(c-1,c-k^2,c),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument, $k$: real or complex modulus and $s$: arclength

Referenced by:

§19.30(i)

Permalink:

http://dlmf.nist.gov/19.30.E3

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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26: 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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19.6.10 $$\underset{\varphi \to 0}{lim}E(\varphi ,k)/\varphi =1.$$

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Symbols:

$E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument and $k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.6.E10

Encodings:

pMML, png, TeX

See also:

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

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27: 19.1 Special Notation

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$l,m,n$	nonnegative integers.
$\varphi$	real or complex argument (or amplitude).
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28: 15.20 Software

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§15.20(ii) Real Parameters and Argument

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29: 4.45 Methods of Computation

… ►The inverses $\mathrm{arcsinh}$, $\mathrm{arccosh}$, and $\mathrm{arctanh}$ can be computed from the logarithmic forms given in §4.37(iv), with real arguments. …

30: 10.40 Asymptotic Expansions for Large Argument

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§10.40(ii) Error Bounds for Real Argument and Order

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