of two complex variables

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1: 1.9 Calculus of a Complex Variable

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Continuity

… ►That is, given any positive number $ϵ$, however small, we can find a positive number $\delta$ such that for all $z$ in the open disk . ►A function of two complex variables $f(z,w)$ is continuous at $(z_0,w_0)$ if $\underset{(z,w)\to (z_0,w_0)}{lim}f(z,w)=f(z_0,w_0)$; compare (1.5.1) and (1.5.2). …

2: 16.13 Appell Functions

… ►The following four functions of two real or complex variables $x$ and $y$ cannot be expressed as a product of two ${}_2{}^{}F_1^{}$ functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): …

3: 19.25 Relations to Other Functions

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19.25.9 $$E(\varphi ,k)=R_F(c-1,c-k^2,c)-\frac{1}{3}{k}^2{R}_D(c-1,c-k^2,c),$$

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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 and $k$: real or complex modulus

Referenced by:

§19.25(i), §19.25(i), §19.25(iii), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.25.E9

Encodings:

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See also:

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

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

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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

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19.25.13 $$D(\varphi ,k)=\frac{1}{3}{R}_D(c-1,c-k^2,c).$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $D(\varphi ,k)$: incomplete elliptic integral of Legendre’s type, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.25.E13

Encodings:

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See also:

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

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19.25.21 $$\mathrm{el2}(x,k_c,a,b)=a\mathrm{el1}(x,k_c)+\frac{1}{3}(b-a)R_D(r,r+k_c^2,r+1),$$

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Symbols:

$\mathrm{el1}(x,k_c)$: Bulirsch’s incomplete elliptic integral of the first kind, $\mathrm{el2}(x,k_c,a,b)$: Bulirsch’s incomplete elliptic integral of the second kind, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $k$: real or complex modulus and $r$: inverse

Permalink:

http://dlmf.nist.gov/19.25.E21

Encodings:

pMML, png, TeX

See also:

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

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19.25.25 $${(z-x)}^{3/2}{R}_D(x,y,z)=(3/k^2)(F(\varphi ,k)-E(\varphi ,k)),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first 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(iii)

Permalink:

http://dlmf.nist.gov/19.25.E25

Encodings:

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See also:

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

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4: Mathematical Introduction

… ►Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. …

5: 31.1 Special Notation

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$x$, $y$	real variables.
$z$, $\zeta$, $w$, $W$	complex variables.
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$a$	complex parameter, $|a|\ge 1,a\ne 1$.
$q,\alpha ,\beta ,\gamma ,\delta ,ϵ,\nu$	complex parameters.

… ►Sometimes the parameters are suppressed.

6: 28.15 Expansions for Small $q$

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28.15.3 $${\mathrm{me}}_{\nu }(z,q)={\mathrm{e}}^{\mathrm{i}\nu z}-\frac{q}{4}\left(\frac{1}{\nu +1}{\mathrm{e}}^{\mathrm{i}(\nu +2)z}-\frac{1}{\nu -1}{\mathrm{e}}^{\mathrm{i}(\nu -2)z}\right)+\frac{q^2}{32}\left(\frac{1}{(\nu +1)(\nu +2)}{\mathrm{e}}^{\mathrm{i}(\nu +4)z}+\frac{1}{(\nu -1)(\nu -2)}{\mathrm{e}}^{\mathrm{i}(\nu -4)z}-\frac{2({\nu }^2+1)}{{({\nu }^2-1)}^2}{\mathrm{e}}^{\mathrm{i}\nu z}\right)+\mathrm{\cdots };$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

20.3.17 (only two terms and without normalization)

Permalink:

http://dlmf.nist.gov/28.15.E3

Encodings:

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See also:

Annotations for §28.15(ii), §28.15 and Ch.28

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7: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

… ►For bounds on $R_n(a,z)$ when $a$ is real and $z$ is complex see Olver (1997b, pp. 109–112). … ►See Tricomi (1950b) for these approximations, together with higher terms and extensions to complex variables. … ►For (8.11.18) and extensions to complex values of $x$ see Buckholtz (1963). …

8: 19.21 Connection Formulas

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19.21.1 $$R_F(0,z+1,z)R_D(0,z+1,1)+R_D(0,z+1,z)R_F(0,z+1,1)=3\pi /(2z),$$ $z\in ℂ\setminus (-\mathrm{\infty },0]$.

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $ℂ$: complex plane, $\in$: element of, $(a,b]$: half-closed interval and $\setminus$: set subtraction

Referenced by:

§19.21(i), §19.21(i), §19.7(i)

Permalink:

http://dlmf.nist.gov/19.21.E1

Encodings:

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See also:

Annotations for §19.21(i), §19.21 and Ch.19

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9: 4.2 Definitions

… ► $\ln z$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },0]$ and real-valued when $z$ ranges over the positive real numbers. … ►In the DLMF we allow a further extension by regarding the cut as representing two sets of points, one set corresponding to the “upper side” and denoted by $z=x+\mathrm{i}0$, the other set corresponding to the “lower side” and denoted by $z=x-\mathrm{i}0$. …Consequently $\ln z$ is two-valued on the cut, and discontinuous across the cut. … ►The function $\exp$ is an entire function of $z$, with no real or complex zeros. … ►This is an analytic function of $z$ on $ℂ\setminus (-\mathrm{\infty },0]$, and is two-valued and discontinuous on the cut shown in Figure 4.2.1, unless $a\in \mathbb{Z}$. …

10: Bibliography B

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Zero-field susceptibility of the two-dimensional Ising model near $T_c$ . Phys. Rev. Lett. 31, pp. 1409–1411.

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External Links:

Document

Referenced by:

§32.16.

Encodings:

BibTeX

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Quantum mechanics of one- and two-electron atoms. Springer-Verlag, Berlin.

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External Links:

MathReview (L. Van Hove), ZentralBlatt

Referenced by:

§18.39(ii), §18.39(ii).

Encodings:

BibTeX

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Quantum Mechanics of One- and Two-electron Atoms. Rosetta edition, Plenum Publishing Corp., New York.

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Notes:

Reprint (with corrections) of the original edition published in 1957 by Springer and Academic Press

External Links:

MathReview, ZentralBlatt (D. Geissler)

Referenced by:

§1.18(viii), §33.22(ii), §33.22(ii), §33.22(v).

Encodings:

BibTeX

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The fifty-two icosahedral solutions to Painlevé VI. J. Reine Angew. Math. 596, pp. 183–214.

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External Links:

ISSN 0075-4102, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.9(iii).

Encodings:

BibTeX

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Several Complex Variables. Princeton Mathematical Series, Vol. 10, Princeton University Press, Princeton, N.J..

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External Links:

MathReview (P. Thullen), ZentralBlatt

Referenced by:

§35.2.

Encodings:

BibTeX

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