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21: 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

22: 23.17 Elementary Properties

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23: 19.26 Addition Theorems

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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

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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

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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

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

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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

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

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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

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24: 13.18 Relations to Other Functions

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25: 8.27 Approximations

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Luke (1975, §4.3) gives Padé approximation methods, combined with a detailed analysis of the error terms, valid for real and complex variables except on the negative real $z$-axis. See also Temme (1994b, §3).

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26: 10.31 Power Series

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27: 10.59 Integrals

… ►Additional integrals can be obtained by combining the definitions (10.47.3)–(10.47.9) with the results given in §10.22 and §10.43. …

28: 19.11 Addition Theorems

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19.11.5 $$\mathrm{\Pi }(\theta ,{\alpha }^2,k)+\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\mathrm{\Pi }(\psi ,{\alpha }^2,k)-{\alpha }^2{R}_C(\gamma -\delta ,\gamma ),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: 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, $\theta$: amplitude, $\psi$: angle, $\gamma$ and $\delta$

Referenced by:

§19.11(i), §19.11(i), Erratum (V1.0.24) for Subsection 19.11(i)

Permalink:

http://dlmf.nist.gov/19.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(i), §19.11 and Ch.19

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19.11.6_5 $$R_C(\gamma -\delta ,\gamma )=\frac{-1}{\sqrt{\delta }}\arctan \left(\frac{\sqrt{\delta }\sin \theta \sin \varphi \sin \psi }{{\alpha }^2-1-{\alpha }^2\cos \theta \cos \varphi \cos \psi }\right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $\arctan z$: arctangent function, $\sin z$: sine function, $\varphi$: real or complex argument, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\psi$: angle, $\gamma$ and $\delta$

Referenced by:

§19.11(i), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)

Permalink:

http://dlmf.nist.gov/19.11.E6_5

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.28):

This equation was given a number.

See also:

Annotations for §19.11(i), §19.11 and Ch.19

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19.11.10 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\mathrm{\Pi }({\alpha }^2,k)-\mathrm{\Pi }(\theta ,{\alpha }^2,k)-{\alpha }^2{R}_C(\gamma -\delta ,\gamma ),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: 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, $\theta$: amplitude, $\gamma$ and $\delta$

Permalink:

http://dlmf.nist.gov/19.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(ii), §19.11 and Ch.19

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19.11.16 $$\mathrm{\Pi }(\psi ,{\alpha }^2,k)=2\mathrm{\Pi }(\theta ,{\alpha }^2,k)+{\alpha }^2{R}_C(\gamma -\delta ,\gamma ),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $k$: real or complex modulus, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\delta$ , $\psi$: angle and $\gamma$

Permalink:

http://dlmf.nist.gov/19.11.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(iii), §19.11 and Ch.19

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29: 9.10 Integrals

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9.10.1 $${\int }_z^{\mathrm{\infty }}\mathrm{Ai}\left(t\right)dt=\pi \left(\mathrm{Ai}\left(z\right){\mathrm{Gi}}^{\prime }\left(z\right)-{\mathrm{Ai}}^{\prime }\left(z\right)\mathrm{Gi}\left(z\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.12.4) and its differentiated form with the first of (9.10.11)

Referenced by:

(9.10.2)

Permalink:

http://dlmf.nist.gov/9.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(i), §9.10 and Ch.9

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9.10.2 $${\int }_{-\mathrm{\infty }}^z\mathrm{Ai}\left(t\right)dt=\pi \left(\mathrm{Ai}\left(z\right){\mathrm{Hi}}^{\prime }\left(z\right)-{\mathrm{Ai}}^{\prime }\left(z\right)\mathrm{Hi}\left(z\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.12.11) and its differentiated form with (9.10.1), then apply (9.2.7) and (9.10.11)

Permalink:

http://dlmf.nist.gov/9.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(i), §9.10 and Ch.9

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9.10.3 $${\int }_{-\mathrm{\infty }}^z\mathrm{Bi}\left(t\right)dt={\int }_0^z\mathrm{Bi}\left(t\right)dt=\pi \left({\mathrm{Bi}}^{\prime }\left(z\right)\mathrm{Gi}\left(z\right)-\mathrm{Bi}\left(z\right){\mathrm{Gi}}^{\prime }\left(z\right)\right)=\pi \left(\mathrm{Bi}\left(z\right){\mathrm{Hi}}^{\prime }\left(z\right)-{\mathrm{Bi}}^{\prime }\left(z\right)\mathrm{Hi}\left(z\right)\right).$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.12.5) and its differentiated form with (9.2.7).

Permalink:

http://dlmf.nist.gov/9.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(i), §9.10 and Ch.9

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9.10.18 $$\mathrm{Ai}\left(z\right)=\frac{3z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{4\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt,$$ .

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Proof sketch:

To prove this equation, combine (13.4.5) with (13.6.11) and use (5.5.3).

Referenced by:

(9.10.19), §9.17(iii), §9.5(ii), Erratum (V1.0.10) for Equation (9.10.18)

Permalink:

http://dlmf.nist.gov/9.10.E18

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

The original and incorrect version,

$$\mathrm{Ai}\left(z\right)=\frac{z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{2^{7/2}\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-1/2}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt$$

taken from Schulten et al. (1979), has been corrected.

Reported 2015-03-20 by Walter Gautschi

See also:

Annotations for §9.10(vii), §9.10 and Ch.9

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9.10.19 $$\mathrm{Bi}\left(x\right)=\frac{3x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt,$$ $x>0$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

Proof sketch:

To prove this equation, combine (9.2.10) with (9.10.18).

Referenced by:

§9.5(i), Erratum (V1.0.10) for Equation (9.10.19)

Permalink:

http://dlmf.nist.gov/9.10.E19

Encodings:

pMML, png, TeX

Errata (effective with 1.0.10):

The original and incorrect version,

$$\mathrm{Bi}\left(x\right)=\frac{x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2^{5/2}\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-1/2}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt$$

taken from Schulten et al. (1979), has been corrected.

Reported 2015-03-20 by Walter Gautschi

See also:

Annotations for §9.10(vii), §9.10 and Ch.9

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30: 19.2 Definitions

… ►Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). … ►

19.2.17 $$R_C(x,y)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{\sqrt{t+x}(t+y)},$$

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Defines:

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

Symbols:

$dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.10(ii)

Permalink:

http://dlmf.nist.gov/19.2.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.18 $$R_C(x,y)=\frac{1}{\sqrt{y-x}}\arctan \sqrt{\frac{y-x}{x}}=\frac{1}{\sqrt{y-x}}\arccos \sqrt{x/y},$$ ,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\arccos z$: arccosine function and $\arctan z$: arctangent function

Referenced by:

§19.2(iv)

Permalink:

http://dlmf.nist.gov/19.2.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.19 $$R_C(x,y)=\frac{1}{\sqrt{x-y}}\mathrm{arctanh}\sqrt{\frac{x-y}{x}}=\frac{1}{\sqrt{x-y}}\ln \frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}},$$ .

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{arctanh}z$: inverse hyperbolic tangent function and $\ln z$: principal branch of logarithm function

Referenced by:

§19.26(i), Figure 19.3.2, Figure 19.3.2, §19.36(ii)

Permalink:

http://dlmf.nist.gov/19.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.21 $$R_C(x,y)={\int }_0^1{(v^{2}x+(1-v^2)y)}^{-1/2}dv,$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§19.2(iv)

Permalink:

http://dlmf.nist.gov/19.2.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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