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21: 1.10 Functions of a Complex Variable

§1.10 Functions of a Complex Variable

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§1.10(i) Taylor’s Theorem for Complex Variables

… ►Let $D$ be a bounded domain with boundary $\partial D$ and let $\overline{D}=D\cup \partial D$. … ►(Or more generally, a simple contour that starts at the center and terminates on the boundary.) … ► …

22: 22.3 Graphics

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§22.3(i) Real Variables: Line Graphs

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Figure 22.3.2: $k=0.7$, $-3K\le x\le 3K$, $K=1.8456\mathrm{\dots }$. For $\mathrm{cn}(x,k)$ the curve for $k=1/\sqrt{2}=0.70710\mathrm{\dots }$ is a boundary between the curves that have an inflection point in the interval $0\le x\le 2K\left(k\right)$, and its translates, and those that do not; see Walker (1996, p. 146). Magnify

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§22.3(ii) Real Variables: Surfaces

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Figure 22.3.15: $\mathrm{dn}(x,k)$ for $k=1-{\mathrm{e}}^{-n}$, $n=0$ to 20, $-5\pi \le x\le 5\pi$. Magnify 3D Help

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§22.3(iii) Complex $z$; Real $k$

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23: Bibliography H

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A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Rational Mech. Anal. 73 (1), pp. 31–51.

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External Links:

ISSN 0003-9527, Document, MathReview (Richard Brown), ZentralBlatt

Referenced by:

§32.11(ii).

Encodings:

BibTeX

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On a Painlevé-type boundary-value problem. Quart. J. Mech. Appl. Math. 37 (4), pp. 525–538.

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External Links:

ISSN 0033-5614, Document, MathReview, ZentralBlatt

Referenced by:

§32.11(i), §32.17.

Encodings:

BibTeX

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Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.

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Notes:

Reprinted in 1979.

External Links:

ISBN 0-8218-1556-3, MathReview, ZentralBlatt

Referenced by:

§35.4(i).

Encodings:

BibTeX

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Variable precision exponential function. ACM Trans. Math. Software 12 (2), pp. 79–91.

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External Links:

ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt

Referenced by:

§4.48(iii).

Encodings:

BibTeX

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24: 28.2 Definitions and Basic Properties

… ►With $\zeta ={\sin }^{2}z$ we obtain the algebraic form of Mathieu’s equation ►

28.2.2 $$\zeta (1-\zeta )w^{\prime \prime }+\frac{1}{2}(1-2\zeta )w^{\prime }+\frac{1}{4}(a-2q(1-2\zeta ))w=0.$$

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Symbols:

$q=h^2$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/28.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(i), §28.2 and Ch.28

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28.2.3 $$(1-{\zeta }^2)w^{\prime \prime }-\zeta w^{\prime }+\left(a+2q-4q{\zeta }^2\right)w=0.$$

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Symbols:

$q=h^2$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution and $\zeta$: change of variable

A&S Ref:

20.1.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(i), §28.2 and Ch.28

… ►Furthermore, a solution $w$ with given initial constant values of $w$ and $w^{\prime }$ at a point $z_0$ is an entire function of the three variables $z$, $a$, and $q$. … ►When $\widehat{\nu }=0$ or $1$, the notation for the two sets of eigenvalues corresponding to each $\widehat{\nu }$ is shown in Table 28.2.1, together with the boundary conditions of the associated eigenvalue problem. …

25: 30.14 Wave Equation in Oblate Spheroidal Coordinates

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§30.14(iv) Separation of Variables

… ►Equation (30.13.7) for $\xi \le {\xi }_0$ together with the boundary condition $w=0$ on the ellipsoid given by $\xi ={\xi }_0$, poses an eigenvalue problem with ${\kappa }^2$ as spectral parameter. …

26: 1.6 Vectors and Vector-Valued Functions

… ►Note: The terminology open and closed sets and boundary points in the $(x,y)$ plane that is used in this subsection and §1.6(v) is analogous to that introduced for the complex plane in §1.9(ii). … ►and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. … ►Suppose $S$ is an oriented surface with boundary $\partial S$ which is oriented so that its direction is clockwise relative to the normals of $S$. … ►Suppose $S$ is a piecewise smooth surface which forms the complete boundary of a bounded closed point set $V$, and $S$ is oriented by its normal being outwards from $V$. …

27: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Ignoring the boundary value terms it follows that … ►Other choices of boundary conditions, identical for $f(x)$ and $g(x)$, and which also lead to the vanishing of the boundary terms in (1.18.26), each lead to a distinct self adjoint extension of $T$. … ►

Self-adjoint extensions of (1.18.28) and the Weyl alternative

… ►Boundary values and boundary conditions for the end point $b$ are defined in a similar way. If $n_1=1$ then there are no nonzero boundary values at $a$; if $n_1=2$ then the above boundary values at $a$ form a two-dimensional class. …

28: Bibliography L

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The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.

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External Links:

ISSN 0002-9947, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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A Numerical Library in Java for Scientists & Engineers. Chapman & Hall/CRC, Boca Raton, FL.

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Notes:

With 1 CD-ROM (Windows, Macintosh and Unix) containing a large general purpose mathematical software collection written in Java including a collection of special function classes. The functions are computed for real variables only. (See Lau (1995) for C version).

External Links:

ISBN 1-58488-430-4, MathReview Entry, ZentralBlatt

Referenced by:

H. T. Lau (1995).

Encodings:

BibTeX

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Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0010-4655, Document, Code, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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Complex Variables. Holden-Day Inc., San Francisco, CA.

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External Links:

MathReview (J. E. Miller), ZentralBlatt

Referenced by:

§1.10(i), §1.10(ii), §1.10(iii), §1.10(vi), §1.10(x), §1.9(i), §1.9(ii), §1.9(iii), §1.9(iv), §1.9(vi), §4.14, §4.2(i), §4.2(iv), §4.20, §4.22, §4.23(ii), §4.23(v), §4.24(ii), §4.28, §4.31, §4.37(ii), §4.4(i), §4.4(ii), §4.7(i), §4.7(ii), §4.8(i), Ch.4.

Encodings:

BibTeX

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Discrete-variable representations and their utilization. In Advances in Chemical Physics, pp. 263–310.

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External Links:

ISBN 9780470141731, Document, Link

Referenced by:

§18.38(i).

Encodings:

BibTeX

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