central in imaginary direction

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21: Annie A. M. Cuyt

… ►She received her Doctorate in Science in 1982 from the same university, summa cum laude and with the felicitations of the jury. …Her main research interest is in the area of numerical approximation theory and its applications to a diversity of problems in scientific computing. …A lot of her research has been devoted to rational approximations, in one as well as in many variables, and sparse interpolation. …She directs the HPC core facility of the University of Antwerp and is a member of the Flemish HPC steering committee. … ►In November 2015, Cuyt was named a Senior Associate Editor of the DLMF.

22: Bernard Deconinck

… ► 1970 in Oudenaarde, Belgium) is a Full Professor in the Department of Applied Mathematics at the University of Washington. He received his Diploma in Engineering Physics from the University of Ghent, Belgium. …in Applied Mathematics from the University of Colorado at Boulder, under the direction of Harvey Segur. In addition, he has spent time at the University of Alberta, the Mathematical Sciences Research Institute in Berkeley, California, and Colorado State University. ►Deconinck is interested in nonlinear waves. …

23: 12.14 The Function $W(a,x)$

… ►In this section solutions of equation (12.2.3) are considered. …In other cases the general theory of (12.2.2) is available. … ►The coefficients $c_{2r}$ and $d_{2r}$ are obtainable by equating real and imaginary parts in … ►As noted in §12.14(ix), when $a$ is negative the solutions of (12.2.3), with $z$ replaced by $x$, are oscillatory on the whole real line; also, when $a$ is positive there is a central interval in which the solutions are exponential in character. In the oscillatory intervals we write …

24: Richard A. Askey

… ► 1933 in St. … ►from Princeton University in 1961 under the direction of Samuel Bochner. …Published in 1985 in the Memoirs of the American Mathematical Society, it also introduced the directed graph of hypergeometric orthogonal polynomials commonly known as the Askey scheme. … Gasper) in 1976. … ► Rankin), published by Academic Press in 1988. …

25: 19.32 Conformal Map onto a Rectangle

… ►

19.32.2 $$dz=-\frac{1}{2}\left(\prod _{j=1}^3{(p-x_j)}^{-1/2}\right)dp,$$ $\mathrm{\Im }p>0$; , $j=1,2,3$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase and $z(p)$: function

Referenced by:

§19.32

Permalink:

http://dlmf.nist.gov/19.32.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.32 and Ch.19

… ►then $z(p)$ is a Schwartz–Christoffel mapping of the open upper-half $p$-plane onto the interior of the rectangle in the $z$-plane with vertices … ►

$$z(x_3)=R_F(x_3-x_1,x_3-x_2,0)=-\mathrm{i}{R}_F(0,x_1-x_3,x_2-x_3).$$

►As $p$ proceeds along the entire real axis with the upper half-plane on the right, $z$ describes the rectangle in the clockwise direction; hence $z(x_3)$ is negative imaginary. …

26: 14.32 Methods of Computation

… ►Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. In particular, for small or moderate values of the parameters $\mu$ and $\nu$ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967). … ►

Application of the uniform asymptotic expansions for large values of the parameters given in §§14.15 and 14.20(vii)–14.20(ix).

… ►

Evaluation (§3.10) of the continued fractions given in §14.14. See Gil and Segura (2000).

…

27: Diego Dominici

… ► 1972 in Buenos Aires, Argentina, d. …in Applied Mathematics at the University of Illinois at Chicago (UIC) under the direction of Charles Knessl and Noemí Irene Wolanski. … ►In 2008 Dominici received a Research Fellowship from the Alexander von Humboldt Foundation and visited the Technische Universität Berlin in Germany. …DiPrima Prize for outstanding research in applied mathematics, awarded by SIAM. ►In November 2015, Dominici was named Associate Editor of the following DLMF Chapters …

28: 18.25 Wilson Class: Definitions

… ►For the Wilson class OP’s $p_n(x)$ with $x=\lambda (y)$: if the $y$-orthogonality set is $\{0,1,\mathrm{\dots },N\}$, then the role of the differentiation operator $d/dx$ in the Jacobi, Laguerre, and Hermite cases is played by the operator ${\mathrm{\Delta }}_y$ followed by division by ${\mathrm{\Delta }}_y(\lambda (y))$, or by the operator ${\nabla }_y$ followed by division by ${\nabla }_y(\lambda (y))$. Alternatively if the $y$-orthogonality interval is $(0,\mathrm{\infty })$, then the role of $d/dx$ is played by the operator ${\delta }_y$ followed by division by ${\delta }_y(\lambda (y))$. … ►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials $W_n(x;a,b,c,d)$, continuous dual Hahn polynomials $S_n(x;a,b,c)$, Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta )$, and dual Hahn polynomials $R_n(x;\gamma ,\delta ,N)$. … ►

18.25.8

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $h_n$

Referenced by:

§18.25(ii)

Permalink:

http://dlmf.nist.gov/18.25.E8

Encodings:

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See also:

Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

… ►

18.25.15 $$h_n=\frac{n!(N-n)!{\left(\gamma +\delta +2\right)}_N}{N!{\left(\gamma +1\right)}_n{\left(\delta +1\right)}_{N-n}}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $N$: positive integer, $\delta$: arbitrary small positive constant, $n$: nonnegative integer and $h_n$

Referenced by:

§18.25(iii)

Permalink:

http://dlmf.nist.gov/18.25.E15

Encodings:

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See also:

Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

…

29: 33.23 Methods of Computation

… ►The methods used for computing the Coulomb functions described below are similar to those in §13.29. … ►When numerical values of the Coulomb functions are available for some radii, their values for other radii may be obtained by direct numerical integration of equations (33.2.1) or (33.14.1), provided that the integration is carried out in a stable direction (§3.7). Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21). … ►In a similar manner to §33.23(iii) the recurrence relations of §§33.4 or 33.17 can be used for a range of values of the integer $\mathrm{\ell }$, provided that the recurrence is carried out in a stable direction (§3.6). …

30: Bibliography W

… ►

Harmonic Analysis. In Studies in Real and Complex Analysis, I. I. Hirschman (Ed.), Studies in Mathematics, Vol. 3, pp. 124–178.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§1.15(iii), §1.15(iv), §1.15(v).

Encodings:

BibTeX

… ►

On the functions associated with the parabolic cylinder in harmonic analysis. Proc. London Math. Soc. 35, pp. 417–427.

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External Links:

Document, ZentralBlatt

Referenced by:

§12.1, §12.5(i), §12.5(ii), §12.9(i).

Encodings:

BibTeX

… ►

The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A, pp. 32–52.

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External Links:

FullText, Document, ZentralBlatt

Referenced by:

§35.3(i), §35.3(ii).

Encodings:

BibTeX

… ►

On the central connection problem for the double confluent Heun equation. Math. Nachr. 195, pp. 267–276.

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External Links:

ISSN 0025-584X, MathReview, ZentralBlatt

Referenced by:

§31.12.

Encodings:

BibTeX

… ►

Applications of some recent results in asymptotic expansions. Congr. Numer. 37, pp. 145–182.

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External Links:

ISSN 0384-9864, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§19.27(vi).

Encodings:

BibTeX

…