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1: 4.3 Graphics

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§4.3(ii) Complex Arguments: Conformal Maps

►Figure 4.3.2 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the negative real axis, where $w={\mathrm{e}}^z$ and $z=\ln w$ (principal value). …Lines parallel to the real axis in the $z$-plane map onto rays in the $w$-plane, and lines parallel to the imaginary axis in the $z$-plane map onto circles centered at the origin in the $w$-plane. … ►

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Figure 4.3.2: Conformal mapping of exponential and logarithm. … Magnify

… ►See also About Color Map. …

2: 15.17 Mathematical Applications

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§15.17(ii) Conformal Mappings

►The quotient of two solutions of (15.10.1) maps the closed upper half-plane $\mathrm{\Im }z\ge 0$ conformally onto a curvilinear triangle. …Hypergeometric functions, especially complete elliptic integrals, also play an important role in quasiconformal mapping. …

3: 4.29 Graphics

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§4.29(ii) Complex Arguments

►The conformal mapping $w=\sinh z$ is obtainable from Figure 4.15.7 by rotating both the $w$-plane and the $z$-plane through an angle $\frac{1}{2}\pi$, compare (4.28.8). …

4: 19.32 Conformal Map onto a Rectangle

§19.32 Conformal Map onto a Rectangle

… ►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 … ►For further connections between elliptic integrals and conformal maps, see Bowman (1953, pp. 44–85).

5: 16.23 Mathematical Applications

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§16.23(iii) Conformal Mapping

►The Bieberbach conjecture states that if ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ is a conformal map of the unit disk to any complex domain, then $|a_n|\le n|a_1|$. …

6: 4.15 Graphics

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§4.15(ii) Complex Arguments: Conformal Maps

►Figure 4.15.7 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the real axis from $-\mathrm{\infty }$ to $-1$ and $1$ to $\mathrm{\infty }$, where $w=\sin z$ and $z=\arcsin w$ (principal value). …Lines parallel to the real axis in the $z$-plane map onto ellipses in the $w$-plane with foci at $w=\pm 1$, and lines parallel to the imaginary axis in the $z$-plane map onto rectangular hyperbolas confocal with the ellipses. … ►

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Figure 4.15.7: Conformal mapping of sine and inverse sine. … Magnify

… ►See also About Color Map. …

7: 23.20 Mathematical Applications

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§23.20(i) Conformal Mappings

… ►The interior of $R$ is mapped one-to-one onto the lower half-plane. … ►For examples of conformal mappings of the function $\mathrm{\wp }\left(z\right)$, see Abramowitz and Stegun (1964, pp. 642–648, 654–655, and 659–60). ►For conformal mappings via modular functions see Apostol (1990, §2.7). …

8: 22.18 Mathematical Applications

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§22.18(ii) Conformal Mapping

►With $k\in [0,1]$ the mapping $z\to w=\mathrm{sn}(z,k)$ gives a conformal map of the closed rectangle $[-K,K]\times [0,K^{\prime }]$ onto the half-plane $\mathrm{\Im }w\ge 0$, with $0,\pm K,\pm K+\mathrm{i}{K}^{\prime },\mathrm{i}{K}^{\prime }$ mapping to $0,\pm 1,\pm k^{-2},\mathrm{\infty }$ respectively. The half-open rectangle $(-K,K)\times [-K^{\prime },K^{\prime }]$ maps onto $ℂ$ cut along the intervals $(-\mathrm{\infty },-1]$ and $[1,\mathrm{\infty })$. See Akhiezer (1990, Chapter 8) and McKean and Moll (1999, Chapter 2) for discussions of the inverse mapping. Bowman (1953, Chapters V–VI) gives an overview of the use of Jacobian elliptic functions in conformal maps for engineering applications. …

9: 29.18 Mathematical Applications

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§29.18(iv) Other Applications

►Triebel (1965) gives applications of Lamé functions to the theory of conformal mappings. …

10: Bibliography

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

Encodings:

BibTeX

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On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

BibTeX

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Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:

ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§10.43(iii).

Encodings:

BibTeX

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Conformal Invariants, Inequalities, and Quasiconformal Maps. John Wiley & Sons Inc., New York.

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Notes:

With PC diskette, a Wiley-Interscience Publication

External Links:

ISBN 0-471-59486-5, MathReview (H. Renelt), ZentralBlatt

Referenced by:

§15.17(ii), §19.9(i).

Encodings:

BibTeX

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Singularities of Differentiable Maps. Vol. II. Birkhäuser, Boston-Berlin.

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Notes:

Monodromy and asymptotics of integrals, Translated from the Russian by Hugh Porteous, Translation revised by the authors and James Montaldi

External Links:

ISBN 0-8176-3185-2, MathReview, ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

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