inverse function

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31: 19.17 Graphics

… ►Because the $R$-function is homogeneous, there is no loss of generality in giving one variable the value $1$ or $-1$ (as in Figure 19.3.2). …The case $y=1$ corresponds to elementary functions. … ►

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Figure 19.17.8: $R_J(0,y,1,p)$, $0\le y\le 1$, $-1\le p\le 2$. …The function is asymptotic to $\frac{3}{2}\pi /\sqrt{yp}$ as $p\to 0+$, and to $(\frac{3}{2}/p)\ln \left(16/y\right)$ as $y\to 0+$. … Magnify 3D Help

32: 22.10 Maclaurin Series

… ►Further terms may be derived from the differential equations (22.13.13), (22.13.14), (22.13.15), or from the integral representations of the inverse functions in §22.15(ii). …

33: 19.12 Asymptotic Approximations

… ►

19.12.4 $$(1-{\alpha }^2)\mathrm{\Pi }({\alpha }^2,k)=\left(\ln \frac{4}{k^{\prime }}\right)\left(1+O\left(k_{}^{\prime }{}_{}{}^2\right)\right)-{\alpha }^2{R}_C(1,1-{\alpha }^2),$$ ,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\ln z$: principal branch of logarithm function, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

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19.12.5 $$(1-{\alpha }^2)\mathrm{\Pi }({\alpha }^2,k)=\left(\ln \left(\frac{4}{k^{\prime }}\right)-R_C(1,1-{\alpha }^{-2})\right)\left(1+O\left(k_{}^{\prime }{}_{}{}^2\right)\right),$$ .

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\ln z$: principal branch of logarithm function, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

… ►

19.12.6 $$R_C(x,y)=\frac{\pi }{2\sqrt{y}}-\frac{\sqrt{x}}{y}\left(1+O\left(\sqrt{\frac{x}{y}}\right)\right),$$ $x/y\to 0$,

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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 and $\pi$: the ratio of the circumference of a circle to its diameter

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

►

19.12.7 $$R_C(x,y)=\frac{1}{2\sqrt{x}}\left(\left(1+\frac{y}{2x}\right)\ln \left(\frac{4x}{y}\right)-\frac{y}{2x}\right)(1+O\left(y^2/x^2\right)),$$ $y/x\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\ln z$: principal branch of logarithm function

Referenced by:

§19.12, §19.2(iv)

Permalink:

http://dlmf.nist.gov/19.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

34: 9.8 Modulus and Phase

… ►

9.8.4 $$\theta \left(x\right)=\arctan \left(\mathrm{Ai}\left(x\right)/\mathrm{Bi}\left(x\right)\right).$$

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Defines:

$\theta \left(z\right)$: Airy phase function

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\arctan z$: arctangent function, $x$: real variable and $z$: complex variable

Source:

Olver (1997b, (2.02), p. 395)

Referenced by:

(9.11.19), (9.8.11), (9.8.15), (9.8.17), (9.8.19)

Permalink:

http://dlmf.nist.gov/9.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

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9.8.8 $$\varphi \left(x\right)=\arctan \left({\mathrm{Ai}}^{\prime }\left(x\right)/{\mathrm{Bi}}^{\prime }\left(x\right)\right).$$

ⓘ

Defines:

$\varphi \left(z\right)$: Airy phase function

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\arctan z$: arctangent function, $x$: real variable and $z$: complex variable

Source:

Olver (1997b, (2.09–2.10), p. 396); with different notation

Referenced by:

(9.11.19), (9.8.12), (9.8.16), (9.8.17)

Permalink:

http://dlmf.nist.gov/9.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

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9.8.11 $$\theta \left(x\right)=\frac{2}{3}\pi +\arctan \left(Y_{1/3}\left(\xi \right)/J_{1/3}\left(\xi \right)\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\theta \left(z\right)$: Airy phase function, $x$: real variable and $\xi$: change of variable

Proof sketch:

Combine definition (9.8.4) with (9.6.17) and (9.6.19) using (10.4.3).

Permalink:

http://dlmf.nist.gov/9.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

►

9.8.12 $$\varphi \left(x\right)=\frac{1}{3}\pi +\arctan \left(Y_{2/3}\left(\xi \right)/J_{2/3}\left(\xi \right)\right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\varphi \left(z\right)$: Airy phase function, $x$: real variable and $\xi$: change of variable

Proof sketch:

Combine definition (9.8.8) with (9.6.18) and (9.6.20) using (10.4.3).

Referenced by:

(9.8.20), (9.8.21), (9.8.22), (9.8.23)

Permalink:

http://dlmf.nist.gov/9.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

…

35: 19.26 Addition Theorems

… ►

19.26.11 $$R_C(x+\lambda ,y+\lambda )+R_C(x+\mu ,y+\mu )=R_C(x,y),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\lambda$: positive, $x$: positive, $y$: positive and $\mu >0$: parameter

Referenced by:

§19.26(i)

Permalink:

http://dlmf.nist.gov/19.26.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(i), §19.26 and Ch.19

… ►

19.26.13 $$R_C({\alpha }^2,{\alpha }^2-\theta )+R_C({\beta }^2,{\beta }^2-\theta )=R_C({\sigma }^2,{\sigma }^2-\theta ),$$ $\sigma =(\alpha \beta +\theta )/(\alpha +\beta )$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.26(i), §19.26(ii)

Permalink:

http://dlmf.nist.gov/19.26.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(i), §19.26 and Ch.19

… ►

19.26.14 $$(p-y)R_C(x,p)+(q-y)R_C(x,q)=(\eta -\xi )R_C(\xi ,\eta ),$$ $x\ge 0$, $y\ge 0$; $p,q\in ℝ\setminus \{0\}$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\in$: element of, $ℝ$: real line, $\setminus$: set subtraction, $x$: positive, $y$: positive, $\xi$: coordinate and $\eta$: coordinate

Referenced by:

§19.26(i)

Permalink:

http://dlmf.nist.gov/19.26.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(i), §19.26 and Ch.19

… ►

19.26.22 $$R_J(x,y,z,p)=2R_J(x+\lambda ,y+\lambda ,z+\lambda ,p+\lambda )+3R_C({\alpha }^2,{\beta }^2),$$

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\lambda$: positive, $\alpha$ , $\beta$ , $x$: positive, $y$: positive and $z$: positive

Referenced by:

§19.26(iii)

Permalink:

http://dlmf.nist.gov/19.26.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(iii), §19.26 and Ch.19

… ►

19.26.25 $$R_C(x,y)=2R_C(x+\lambda ,y+\lambda ),$$ $\lambda =y+2\sqrt{x}\sqrt{y}$.

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\lambda$: positive, $x$: positive and $y$: positive

Referenced by:

§19.26(iii)

Permalink:

http://dlmf.nist.gov/19.26.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(iii), §19.26 and Ch.19

…

36: 19.8 Quadratic Transformations

… ►

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

ⓘ

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

… ►

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

ⓘ

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

…

37: 12.11 Zeros

… ►where $t(\zeta )$ is the function inverse to $\zeta (t)$, defined by (12.10.39) (see also (12.10.41)), and …

38: 14.15 Uniform Asymptotic Approximations

… ►where the inverse trigonometric functions take their principal values (§4.23(ii)). … ►

14.15.27 $$\frac{1}{2}\zeta {\left({\zeta }^2-{\alpha }^2\right)}^{1/2}-\frac{1}{2}{\alpha }^2\mathrm{arccosh}\left(\frac{\zeta }{\alpha }\right)={\left(1-a^2\right)}^{1/2}\mathrm{arctanh}\left(\frac{1}{x}{\left(\frac{x^2-a^2}{1-a^2}\right)}^{1/2}\right)-\mathrm{arccosh}\left(\frac{x}{a}\right),$$ , ,

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $x$: real variable, $a$ , $\zeta$ and $\alpha$

Permalink:

http://dlmf.nist.gov/14.15.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(v), §14.15 and Ch.14

… ►The inverse hyperbolic and trigonometric functions take their principal values (§§4.23(ii), 4.37(ii)). … ►

14.15.31 $$\frac{1}{2}\zeta {\left({\zeta }^2+{\alpha }^2\right)}^{1/2}+\frac{1}{2}{\alpha }^2\mathrm{arcsinh}\left(\frac{\zeta }{\alpha }\right)={\left(1+a^2\right)}^{1/2}\mathrm{arctanh}\left(x{\left(\frac{1+a^2}{x^2+a^2}\right)}^{1/2}\right)-\mathrm{arcsinh}\left(\frac{x}{a}\right),$$ , ,

ⓘ

Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $x$: real variable, $a$ , $\zeta$ and $\alpha$

Permalink:

http://dlmf.nist.gov/14.15.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(v), §14.15 and Ch.14

… ►(The inverse hyperbolic functions again take their principal values.) …

39: 19.20 Special Cases

… ►

19.20.5 $$2R_G(x,y,y)=y{R}_C(x,y)+\sqrt{x}.$$

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_G(x,y,z)$: symmetric elliptic integral of second kind

Referenced by:

§19.20(ii)

Permalink:

http://dlmf.nist.gov/19.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(ii), §19.20 and Ch.19

… ►

19.20.13 $$2(p-x)R_J(x,y,z,p)=3R_F(x,y,z)-3\sqrt{x}{R}_C(yz,p^2),$$ $p=x\pm \sqrt{(y-x)(z-x)}$,

ⓘ

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 and $R_J(x,y,z,p)$: symmetric elliptic integral of third kind

Referenced by:

§19.20(iii), §19.22(iii)

Permalink:

http://dlmf.nist.gov/19.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iii), §19.20 and Ch.19

… ►When the variables are real and distinct, the various cases of $R_J(x,y,z,p)$ are called circular (hyperbolic) cases if $(p-x)(p-y)(p-z)$ is positive (negative), because they typically occur in conjunction with inverse circular (hyperbolic) functions. … ►

19.20.20 $$R_D(x,y,y)=\frac{3}{2(y-x)}\left(R_C(x,y)-\frac{\sqrt{x}}{y}\right),$$ $x\ne y$, $y\ne 0$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.20(iv), §19.22(iii)

Permalink:

http://dlmf.nist.gov/19.20.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

►

19.20.21 $$R_D(x,x,z)=\frac{3}{z-x}\left(R_C(z,x)-\frac{1}{\sqrt{z}}\right),$$ $x\ne z$, $xz\ne 0$.

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.20.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

…

40: 19.30 Lengths of Plane Curves

… ►

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

ⓘ

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

…