of multivalued function

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31—37 of 37 matching pages

31: 7.18 Repeated Integrals of the Complementary Error Function

… ►The confluent hypergeometric function on the right-hand side of (7.18.10) is multivalued and in the sectors

\tfrac{1}{2}\pi<\left|\operatorname{ph}z\right|<\pi

one has to use the analytic continuation formula (13.2.12). …

32: 12.7 Relations to Other Functions

… ►(It should be observed that the functions on the right-hand sides of (12.7.14) are multivalued; hence, for example,

z

cannot be replaced simply by

-z

33: 19.18 Derivatives and Differential Equations

… ►

§19.18(i) Derivatives

… ►

§19.18(ii) Differential Equations

… ►The function

w=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};x+y,x-y\right)

satisfies an Euler–Poisson–Darboux equation: …Also

W=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};t+r,t-r\right)

, with

r=\sqrt{x^{2}+y^{2}}

, satisfies a wave equation: …Similarly, the function

u=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};x+iy,x-iy\right)

satisfies an equation of axially symmetric potential theory: …

34: 19.2 Definitions

… ►Let

s^{2}(t)

be a cubic or quartic polynomial in

t

with simple zeros, and let

r(s,t)

be a rational function of

s

and

t

containing at least one odd power of

s

. …where

p_{j}

is a polynomial in

t

while

\rho

and

\sigma

are rational functions of

t

. … ►

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

… ►In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). When

x

and

y

are positive,

R_{C}\left(x,y\right)

is an inverse circular function if

x<y

and an inverse hyperbolic function (or logarithm) if

x>y

: …

35: 19.20 Special Cases

… ►In this subsection, and also §§19.20(ii)–19.20(v), the variables of all

R

-functions satisfy the constraints specified in §19.16(i) unless other conditions are stated. … ►

19.20.5

2R_{G}\left(x,y,y\right)=yR_{C}\left(x,y\right)+\sqrt{x}.

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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

… ►When the variables are real and distinct, the various cases of

R_{J}\left(x,y,z,p\right)

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. … ►

19.20.19

R_{D}\left(x,y,z\right)\sim 3x^{-1/2}y^{-1/2}z^{-1/2},

z/\sqrt{xy}\to 0

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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

… ►

19.20.25

R_{-c}\left(\mathbf{b};\mathbf{z}\right)=\prod_{j=1}^{n}z_{j}^{-b_{j}},

ⓘ

Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function and $n$ : nonnegative integer
Referenced by:: §19.18(ii), §19.21(ii)
Permalink:: http://dlmf.nist.gov/19.20.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §19.20(v), §19.20 and Ch.19

…

36: 19.25 Relations to Other Functions

§19.25 Relations to Other Functions

… ►

§19.25(iv) Theta Functions

… ►

§19.25(v) Jacobian Elliptic Functions

… ► ►

§19.25(vi) Weierstrass Elliptic Functions

…

37: 19.21 Connection Formulas

… ►The complete cases of

R_{F}

and

R_{G}

have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … ►

19.21.7

(x-y)R_{D}\left(y,z,x\right)+(z-y)R_{D}\left(x,y,z\right)=3R_{F}\left(x,y,z% \right)-3y^{1/2}x^{-1/2}z^{-1/2},

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.21(ii), §19.25(i), Erratum (V1.1.3) for Chapter 19
Permalink:: http://dlmf.nist.gov/19.21.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 §19.21(ii), §19.21 and Ch.19

… ►

19.21.8

R_{D}\left(y,z,x\right)+R_{D}\left(z,x,y\right)+R_{D}\left(x,y,z\right)=3x^{-1% /2}y^{-1/2}z^{-1/2},

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables
Referenced by:: §19.21(ii), §19.33(iii), Erratum (V1.1.3) for Chapter 19
Permalink:: http://dlmf.nist.gov/19.21.E8
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 §19.21(ii), §19.21 and Ch.19

… ►

19.21.10

2R_{G}\left(x,y,z\right)=zR_{F}\left(x,y,z\right)-\tfrac{1}{3}(x-z)(y-z)R_{D}% \left(x,y,z\right)+x^{1/2}y^{1/2}z^{-1/2},

z\neq 0

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.20(ii), §19.21(i), §19.21(ii), §19.21(ii), §19.36(i), §19.36(ii), Erratum (V1.1.3) for Chapter 19, Erratum (V1.1.7) for Equation (19.21.10)
Permalink:: http://dlmf.nist.gov/19.21.E10
Encodings:: pMML, png, TeX
Correction (effective with 1.1.7):: An incorrect factor of 3 was removed in the last term on the right-hand side.
Suggested 2022-06-27 by Abdulhafeez Abdulsalam
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 §19.21(ii), §19.21 and Ch.19

… ►

19.21.12

(p-x)R_{J}\left(x,y,z,p\right)+(q-x)R_{J}\left(x,y,z,q\right)=3R_{F}\left(x,y,% z\right)-3R_{C}\left(\xi,\eta\right),

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\xi$ and $\eta$
Referenced by:: §19.15, §19.20(iii), §19.21(iii), §19.25(i), §19.26(i)
Permalink:: http://dlmf.nist.gov/19.21.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(iii), §19.21 and Ch.19

…