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21: 3.5 Quadrature

… ► … ►a complex Gauss quadrature formula is available. Here $f(\zeta )$ is assumed analytic in the half-plane $\mathrm{\Re }\zeta >c_0$ and bounded as $\zeta \to \mathrm{\infty }$ in $\left|\mathrm{ph}\zeta \right|\le \frac{1}{2}\pi$. … … ►The integrand can be extended as a periodic $C^{\mathrm{\infty }}$ function on $ℝ$ with period $2\pi$ and as noted in §3.5(i), the trapezoidal rule is exceptionally efficient in this case. …

22: Mathematical Introduction

… ►This process greatly extended normal editorial checking procedures. … ►(These chapters can also serve as background material for university graduate courses in complex variables, classical analysis, and numerical analysis.) … ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). … ►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. …However, in many cases the coloring of the surface is chosen instead to indicate the quadrant of the plane to which the phase of the function belongs, thereby achieving a 4D effect. …

23: 2.7 Differential Equations

… ►The first of these references includes extensions to complex variables and reversions for zeros. … ►In consequence, if a differential equation has more than one singularity in the extended plane, then usually more than two standard solutions need to be chosen in order to have numerically satisfactory representations everywhere. …

24: 8.21 Generalized Sine and Cosine Integrals

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8.21.11 $$\mathrm{Si}(0,z)=\mathrm{Si}\left(z\right).$$

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Symbols:

$\mathrm{Si}(a,z)$: generalized sine integral, $\mathrm{Si}\left(z\right)$: sine integral and $z$: complex variable

Referenced by:

§8.21(v)

Permalink:

http://dlmf.nist.gov/8.21.E11

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See also:

Annotations for §8.21(v), §8.21 and Ch.8

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8.21.20 $$\mathrm{si}(a,z)=f(a,z)\cos z+g(a,z)\sin z,$$

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Symbols:

$\cos z$: cosine function, $\mathrm{si}(a,z)$: generalized sine integral, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $f(a,z)$: auxiliary function and $g(a,z)$: auxiliary function

Referenced by:

§8.21(viii)

Permalink:

http://dlmf.nist.gov/8.21.E20

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Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.21 $$\mathrm{ci}(a,z)=-f(a,z)\sin z+g(a,z)\cos z.$$

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Symbols:

$\cos z$: cosine function, $\mathrm{ci}(a,z)$: generalized cosine integral, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $f(a,z)$: auxiliary function and $g(a,z)$: auxiliary function

Referenced by:

§8.21(viii)

Permalink:

http://dlmf.nist.gov/8.21.E21

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See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.22 $$f(a,z)={\int }_0^{\mathrm{\infty }}\frac{\sin t}{{(t+z)}^{1-a}}dt,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $z$: complex variable, $a$: parameter and $f(a,z)$: auxiliary function

A&S Ref:

5.2.12 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E22

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Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.23 $$g(a,z)={\int }_0^{\mathrm{\infty }}\frac{\cos t}{{(t+z)}^{1-a}}dt.$$

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable, $a$: parameter and $g(a,z)$: auxiliary function

A&S Ref:

5.2.13 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E23

Encodings:

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See also:

Annotations for §8.21(vii), §8.21 and Ch.8

…

25: 2.4 Contour Integrals

… ►The result in §2.3(ii) carries over to a complex parameter $z$. Except that $\lambda$ is now permitted to be complex, with $\mathrm{\Re }\lambda >0$, we assume the same conditions on $q(t)$ and also that the Laplace transform in (2.3.8) converges for all sufficiently large values of $\mathrm{\Re }z$. … ►(The branches of $t^{(s+\lambda -\mu )/\mu }$ and $z^{(s+\lambda )/\mu }$ are extended by continuity.) … ►in the half-plane $\mathrm{\Re }z\ge c$. … ►Zeros of $p^{\prime }(t)$ are called saddle points (or cols) owing to the shape of the surface $|p(t)|$, $t\in ℂ$, in their vicinity. …

26: Bibliography R

… ►

Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric. IEEE Trans. Antennas and Propagation 39 (2), pp. 218–223.

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External Links:

Document

Referenced by:

6th item.

Encodings:

BibTeX

… ►

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Notes:

Symbolic and extended-precision general purpose computer algebra system.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

Encodings:

BibTeX

… ►

Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

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External Links:

MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

Encodings:

BibTeX

… ►

Diffraction of plane radio waves by a parabolic cylinder. Calculation of shadows behind hills. Bell System Tech. J. 33, pp. 417–504.

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External Links:

MathReview (J. Shmoys)

Referenced by:

§12.17.

Encodings:

BibTeX

… ►

Real and complex analysis. McGraw-Hill Book Co., New York-Toronto, Ont.-London.

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External Links:

MathReview (R. E. Edwards)

Referenced by:

§1.2(v), §1.4(v), §1.5(v).

Encodings:

BibTeX

…

27: 2.6 Distributional Methods

… ►The distribution method outlined here can be extended readily to functions $f(t)$ having an asymptotic expansion of the form … ►Furthermore, $K^{+}$ contains the distributions $H$, $\delta$, and $t^{\lambda }$, $t>0$, for any real (or complex) number $\lambda$, where $H$ is the distribution associated with the Heaviside function $H\left(t\right)$ (§1.16(iv)), and $t^{\lambda }$ is the distribution defined by (2.6.12)–(2.6.14), depending on the value of $\lambda$. … ►The method of distributions can be further extended to derive asymptotic expansions for convolution integrals: … ►

2.6.59 $${\int }_0^{\mathrm{\infty }}{t}^{\lambda }dt=0,$$ $\lambda \in ℂ$.

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Symbols:

$ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of and $\int$: integral

Referenced by:

§2.6(iv)

Permalink:

http://dlmf.nist.gov/2.6.E59

Encodings:

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See also:

Annotations for §2.6(iv), §2.6 and Ch.2

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28: 18.38 Mathematical Applications

… ►

Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

… ►Light and Carrington Jr. (2000) review and extend the one-dimensional analysis to solution of multi-dimensional many-particle systems, where the sparse nature of the resulting matrices is highly advantageous. … ►

Complex Function Theory

… ►also the case $\beta =0$ of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. … ►These generalize the ladder operators, as reviewed and extended by Infeld and Hull (1951), and also called creation and annilhilation operators. …