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31: 19.20 Special Cases

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

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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

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

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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

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19.20.14 $$(q+z)R_J(x,y,z,-q)=(p-z)R_J(x,y,z,p)-3R_F(x,y,z)+3{\left(\frac{xyz}{xy+pq}\right)}^{1/2}{R}_C(xy+pq,pq),$$

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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:

§19.16(i), §19.16(i), §19.20(iii), §19.20(iii), §19.21(iii), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.20.E14

Encodings:

pMML, png, TeX

See also:

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

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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

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

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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)

Permalink:

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

Encodings:

pMML, png, TeX

See also:

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

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32: 10.38 Derivatives with Respect to Order

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10.38.1 $$\frac{\partial I_{\pm \nu }\left(z\right)}{\partial \nu }=\pm I_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.42

Referenced by:

§10.38, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.38.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial I_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.38.2).

See also:

Annotations for §10.38 and Ch.10

… ►For $\partial I_{\nu }\left(z\right)/\partial \nu$ at $\nu =-n$ combine (10.38.1), (10.38.2), and (10.38.4). …

33: 10.8 Power Series

… ►When $\nu$ is not an integer the corresponding expansions for $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, and $H_{\nu }^{(2)}\left(z\right)$ are obtained by combining (10.2.2) with (10.2.3), (10.4.7), and (10.4.8). …

34: 10.29 Recurrence Relations and Derivatives

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35: 26.1 Special Notation

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$x$	real variable.
…

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36: 8.17 Incomplete Beta Functions

… ►Further integral representations can be obtained by combining the results given in §8.17(ii) with §15.6. …

37: 10.65 Power Series

… ►When $\nu$ is not an integer combine (10.65.1) with (10.61.6). …

38: 19.27 Asymptotic Approximations and Expansions

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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

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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

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

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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, $b$ 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

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39: 13.8 Asymptotic Approximations for Large Parameters

… ►When the foregoing results are combined with Kummer’s transformation (13.2.39), an approximation is obtained for the case when $|b|$ is large, and $|b-a|$ and $|z|$ are bounded. … ►To obtain approximations for $M(a,b,z)$ and $U(a,b,z)$ that hold as $b\to \mathrm{\infty }$, with $a>\frac{1}{2}-b$ and $z>0$ combine (13.14.4), (13.14.5) with §13.20(i). ►Also, more complicated—but more powerful—uniform asymptotic approximations can be obtained by combining (13.14.4), (13.14.5) with §§13.20(iii) and 13.20(iv). … ►For asymptotic approximations to $M(a,b,x)$ and $U(a,b,x)$ as $a\to -\mathrm{\infty }$ that hold uniformly with respect to $x\in (0,\mathrm{\infty })$ and bounded positive values of $(b-1)/\left|a\right|$, combine (13.14.4), (13.14.5) with §§13.21(ii), 13.21(iii). … ►

13.8.16 $$(k+1)c_{k+1}(z)+\sum _{s=0}^k\left(\frac{b{B}_{s+1}}{(s+1)!}+\frac{z(s+1)B_{s+2}}{(s+2)!}\right)c_{k-s}(z)=0,$$ $k=0,1,2,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.8.16), §13.8(iii)

Permalink:

http://dlmf.nist.gov/13.8.E16

Encodings:

pMML, png, TeX

Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added. In Temme (2015, Equation 10.3.28) the generating function $f(z,t)$ for $c_k(z)$ is given. It satisfies,

$$\frac{\partial f}{\partial t}=\left(b\left(\frac{1}{t}-\frac{1}{{\mathrm{e}}^t-1}\right)-z\left(\frac{1}{t^2}-\frac{{\mathrm{e}}^t}{{\left({\mathrm{e}}^t-1\right)}^2}\right)\right)f.$$

For (13.8.16) combine this differential equation with $f(z,t)={\sum }_{k=0}^{\mathrm{\infty }}{c}_k(z)t^k$ and (24.4.1).

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

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40: 10.49 Explicit Formulas

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