inverse function

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41: 16.15 Integral Representations and Integrals

… ►For inverse Laplace transforms of Appell functions see Prudnikov et al. (1992b, §3.40).

42: 19.27 Asymptotic Approximations and Expansions

… ►

19.27.13 $$R_J(x,y,z,p)=\frac{3}{2\sqrt{z}p}\left(\ln \left(\frac{8z}{a+g}\right)-2R_C(1,\frac{p}{z})+O\left(\left(\frac{a}{z}+\frac{a}{p}\right)\ln \frac{p}{a}\right)\right),$$ $\max (x,y)/\min (z,p)\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.12, §19.20(iii)

Permalink:

http://dlmf.nist.gov/19.27.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(v), §19.27 and Ch.19

►

19.27.14 $$R_J(x,y,z,p)=\frac{3}{\sqrt{yz}}{R}_C(x,p)-\frac{6}{yz}{R}_G(0,y,z)+O\left(\frac{\sqrt{x+2p}}{yz}\right),$$ $\max (x,p)/\min (y,z)\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind

Referenced by:

§19.20(iii)

Permalink:

http://dlmf.nist.gov/19.27.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(v), §19.27 and Ch.19

… ►

19.27.16 $$R_J(x,y,z,p)=(3/\sqrt{x})R_C({(h+p)}^2,2(b+h)p)+O\left(\frac{1}{x^{3/2}}\ln \frac{x}{b+h}\right),$$ $\max (y,z,p)/x\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\ln z$: principal branch of logarithm function, $b$ and $h$

Permalink:

http://dlmf.nist.gov/19.27.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(v), §19.27 and Ch.19

…

43: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

… ►

13.21.12 $$\sqrt{\kappa \zeta -4{\mu }^2}-2\mu \arctan \left(\frac{\sqrt{\kappa \zeta -4{\mu }^2}}{2\mu }\right)=\frac{1}{2}(X-\pi \mu )-\mu \arctan \left(\frac{x\kappa -2{\mu }^2}{\mu X}\right)+\kappa \arcsin \left(\frac{X}{2\sqrt{{\kappa }^2-{\mu }^2}}\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arcsin z$: arcsine function, $\arctan z$: arctangent function, $x$: real variable and $X$

Permalink:

http://dlmf.nist.gov/13.21.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(ii), §13.21 and Ch.13

… ►

13.21.20 $$\widehat{\zeta }=-{\left(\frac{3}{2\kappa }\left(-\frac{1}{2}X+2\mu \arctan \left(\frac{x\kappa -x\sqrt{{\kappa }^2-{\mu }^2}-2{\mu }^2}{\mu X}\right)+\kappa \arccos \left(\frac{x-2\kappa }{2\sqrt{{\kappa }^2-{\mu }^2}}\right)\right)\right)}^{2/3},$$ ,

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Symbols:

$\arccos z$: arccosine function, $\arctan z$: arctangent function, $x$: real variable and $X$

Permalink:

http://dlmf.nist.gov/13.21.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(iii), §13.21 and Ch.13

…

44: 13.20 Uniform Asymptotic Approximations for Large $\mu$

… ►

13.20.9 $$\zeta \sqrt{{\zeta }^2+{\alpha }^2}+{\alpha }^2\mathrm{arcsinh}\left(\frac{\zeta }{\alpha }\right)=\frac{X}{\mu }-\frac{2\kappa }{\mu }\ln \left(\frac{X+x-2\kappa }{2\sqrt{{\mu }^2-{\kappa }^2}}\right)-2\ln \left(\frac{\mu X+2{\mu }^2-\kappa x}{x\sqrt{{\mu }^2-{\kappa }^2}}\right).$$

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Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iii), §13.20 and Ch.13

… ►

13.20.13 $$\zeta \sqrt{{\zeta }^2-{\alpha }^2}-{\alpha }^2\mathrm{arccosh}\left(\frac{\zeta }{\alpha }\right)=\frac{X}{\mu }-\frac{2\kappa }{\mu }\ln \left(\frac{X+x-2\kappa }{2\sqrt{{\kappa }^2-{\mu }^2}}\right)-2\ln \left(\frac{\kappa x-\mu X-2{\mu }^2}{x\sqrt{{\kappa }^2-{\mu }^2}}\right),$$ $x\ge 2\kappa +2\sqrt{{\kappa }^2-{\mu }^2}$,

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

►

13.20.14 $$\zeta \sqrt{{\alpha }^2-{\zeta }^2}+{\alpha }^2\arcsin \left(\frac{\zeta }{\alpha }\right)=\frac{X}{\mu }+\frac{2\kappa }{\mu }\arctan \left(\frac{x-2\kappa }{X}\right)-2\arctan \left(\frac{\kappa x-2{\mu }^2}{\mu X}\right),$$ $2\kappa -2\sqrt{{\kappa }^2-{\mu }^2}\le x\le 2\kappa +2\sqrt{{\kappa }^2-{\mu }^2}$,

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Symbols:

$\arcsin z$: arcsine function, $\arctan z$: arctangent function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

►

13.20.15 $$-\zeta \sqrt{{\zeta }^2-{\alpha }^2}-{\alpha }^2\mathrm{arccosh}\left(-\frac{\zeta }{\alpha }\right)=-\frac{X}{\mu }+\frac{2\kappa }{\mu }\ln \left(\frac{2\kappa -X-x}{2\sqrt{{\kappa }^2-{\mu }^2}}\right)+2\ln \left(\frac{\mu X+2{\mu }^2-\kappa x}{x\sqrt{{\kappa }^2-{\mu }^2}}\right),$$ ,

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\alpha$ and $X$

Permalink:

http://dlmf.nist.gov/13.20.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iv), §13.20 and Ch.13

…

45: 20.9 Relations to Other Functions

… ►The relations (20.9.1) and (20.9.2) between $k$ and $\tau$ (or $q$) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …

46: 22.14 Integrals

… ►

22.14.2 $$\int \mathrm{cn}(x,k)dx=k^{-1}\mathrm{Arccos}\left(\mathrm{dn}(x,k)\right),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arccos}z$: general arccosine function, $x$: real and $k$: modulus

A&S Ref:

16.24.2

Permalink:

http://dlmf.nist.gov/22.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

►

22.14.3 $$\int \mathrm{dn}(x,k)dx=\mathrm{Arcsin}\left(\mathrm{sn}(x,k)\right)=\mathrm{am}(x,k).$$

ⓘ

Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arcsin}z$: general arcsine function, $x$: real and $k$: modulus

A&S Ref:

16.24.3

Permalink:

http://dlmf.nist.gov/22.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

►The branches of the inverse trigonometric functions are chosen so that they are continuous. … ►

22.14.5 $$\int \mathrm{sd}(x,k)dx={(k{k}^{\prime })}^{-1}\mathrm{Arcsin}\left(-k\mathrm{cd}(x,k)\right),$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arcsin}z$: general arcsine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.24.5

Permalink:

http://dlmf.nist.gov/22.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

… ►Again, the branches of the inverse trigonometric functions must be continuous. …

47: 15.4 Special Cases

… ►

15.4.3 $$F(\frac{1}{2},1;\frac{3}{2};-z^2)=z^{-1}\arctan z,$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\arctan z$: arctangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/15.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(i), §15.4 and Ch.15

►

15.4.4 $$F(\frac{1}{2},\frac{1}{2};\frac{3}{2};z^2)=z^{-1}\arcsin z,$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\arcsin z$: arcsine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/15.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(i), §15.4 and Ch.15

…

48: 19.7 Connection Formulas

… ►

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

… ►

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

ⓘ

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

… ►

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

ⓘ

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

49: Guide to Searching the DLMF

… ►

All the inverse trigonometric functions (arcsin vs. Arcsin, etc.).

…

50: 19.23 Integral Representations

… ►

19.23.4 $$R_F(0,y,z)=\frac{2}{\pi }{\int }_0^{\pi /2}{R}_C(y,z{\cos }^2\theta )d\theta =\frac{2}{\pi }{\int }_0^{\mathrm{\infty }}{R}_C(y{\cosh }^{2}t,z)dt.$$

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function and $\int$: integral

Referenced by:

§19.23

Permalink:

http://dlmf.nist.gov/19.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

►

19.23.5 $$R_F(x,y,z)=\frac{2}{\pi }{\int }_0^{\pi /2}{R}_C(x,y{\cos }^2\theta +z{\sin }^2\theta )d\theta ,$$ $\mathrm{\Re }y>0$, $\mathrm{\Re }z>0$,

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\sin z$: sine function

Referenced by:

§19.2(iv), §19.23

Permalink:

http://dlmf.nist.gov/19.23.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.23 and Ch.19

…