multivalued

SearchAdvancedHelp

(0.000 seconds)

31—37 of 37 matching pages

31: 19.18 Derivatives and Differential Equations

… ►

19.18.6 $$\left(\frac{\partial }{\partial x}+\frac{\partial }{\partial y}+\frac{\partial }{\partial z}\right)R_F(x,y,z)=\frac{-1}{2\sqrt{x}\sqrt{y}\sqrt{z}},$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Referenced by:

§19.18(ii), §19.32, Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.18.E6

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

…

32: 22.15 Inverse Functions

… ►Each of these inverse functions is multivalued. …

33: Errata

… ►

Factors inside square roots on the right-hand sides of formulas (19.18.6), (19.20.10), (19.20.19), (19.21.7), (19.21.8), (19.21.10), (19.25.7), (19.25.10) and (19.25.11) were written as products to ensure the correct multivalued behavior.

Reported by Luc Maisonobe on 2021-06-07

… ►

A note about the multivalued nature of the Kummer confluent hypergeometric function of the second kind $U$ on the right-hand side of (7.18.10) was inserted.

… ►

It was reported by Nico Temme on 2015-02-28 that the asymptotic formula for $\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)$ is valid for $h$ $(\in ℂ)$; originally it was unnecessarily restricted to $[0,1]$.

… ►

To increase the regions of validity the logarithms of the gamma function that appears on their left-hand sides have all been changed to $\mathrm{Ln}\mathrm{\Gamma }(\cdot )$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }(\cdot )$ was used, where $\ln$ is the principal branch of the logarithm. These changes were recommended by Philippe Spindel on 2015-02-06.

… ►

5.17.5 $$\mathrm{Ln}G\left(z+1\right)\sim \frac{1}{4}{z}^2+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-\left(\frac{1}{2}z(z+1)+\frac{1}{12}\right)\mathrm{Ln}z-\ln A+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}$$

Originally the term $z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)$ was incorrectly stated as $z\mathrm{\Gamma }\left(z+1\right)$.

Reported 2013-08-01 by Gergő Nemes and subsequently by Nick Jones on December 11, 2013.

…

34: 19.20 Special Cases

… ►

19.20.19 $$R_D(x,y,z)\sim 3x^{-1/2}{y}^{-1/2}{z}^{-1/2},$$ $z/\sqrt{xy}\to 0$.

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables and $\sim$: asymptotic equality

Referenced by:

§19.20(iv), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.20.E19

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

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

…

35: 19.25 Relations to Other Functions

… ►

19.25.7 $$E(\varphi ,k)=2R_G(c-1,c-k^2,c)-(c-1)R_F(c-1,c-k^2,c)-\sqrt{c-1}\sqrt{c-k^2}/\sqrt{c},$$

ⓘ

Symbols:

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

Referenced by:

§19.25(i), §19.25(i), §19.25(iii), Figure 19.3.4, Figure 19.3.4, Figure 19.3.4, §19.36(ii), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.25.E7

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.25(i), §19.25 and Ch.19

… ►

19.25.10 $$E(\varphi ,k)=k_{}^{\prime }{}_{}{}^2{R}_F(c-1,c-k^2,c)+\frac{1}{3}{k}^{2}k_{}^{\prime }{}_{}{}^2{R}_D(c-1,c,c-k^2)+k^2\sqrt{c-1}/\left(\sqrt{c}\sqrt{c-k^2}\right),$$ $c>k^2$,

ⓘ

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 $k^{\prime }$: complementary modulus

Referenced by:

§19.25(i), §19.30(i), §19.36(i), §19.6(iv), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.25.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.25(i), §19.25 and Ch.19

►

19.25.11 $$E(\varphi ,k)=-\frac{1}{3}k_{}^{\prime }{}_{}{}^2{R}_D(c-k^2,c,c-1)+\sqrt{c-k^2}/\left(\sqrt{c}\sqrt{c-1}\right),$$ $\varphi \ne \frac{1}{2}\pi$.

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\pi$: the ratio of the circumference of a circle to its diameter, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.25(i), §19.25(i), §19.36(i), §19.6(iv), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.25.E11

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.25(i), §19.25 and Ch.19

…

36: 8.21 Generalized Sine and Cosine Integrals

… ►From §§8.2(i) and 8.2(ii) it follows that each of the four functions $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ is a multivalued function of $z$ with branch point at $z=0$. …

37: 19.2 Definitions

…