conformal mapping

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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). … ►

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

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2: 15.17 Mathematical Applications

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§15.17(ii) Conformal Mappings

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3: 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 $|a_n|\le n|a_1|$. …

4: 19.32 Conformal Map onto a Rectangle

§19.32 Conformal Map onto a Rectangle

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

5: 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). …

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). … ►

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

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7: 23.20 Mathematical Applications

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§23.20(i) Conformal Mappings

… ►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. …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 H

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Applied and Computational Complex Analysis. Vol. 1: Power Series—Integration—Conformal Mapping—Location of Zeros. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York.

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Notes:

Reprinted in 1988.

External Links:

ISBN 0-471-37244-7, MathReview (M. Marden), ZentralBlatt

Referenced by:

§1.10(vii), §1.9(vi), Ch.3.

Encodings:

BibTeX

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Applied and Computational Complex Analysis. Vol. 3: Discrete Fourier Analysis—Cauchy Integrals—Construction of Conformal Maps—Univalent Functions. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons Inc.], New York.

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Notes:

Reprinted in 1993.

External Links:

ISBN 0-471-08703-3; 0-471-58986-1, MathReview (A. E. Heins), ZentralBlatt

Referenced by:

§1.14(v), Ch.3.

Encodings:

BibTeX

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