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1: 4.3 Graphics
§4.3(ii) Complex Arguments: Conformal Maps
Figure 4.3.2 illustrates the conformal mapping of the strip - π < z < π onto the whole w -plane cut along the negative real axis, where w = e z and z = ln w (principal value). …
See accompanying text
Figure 4.3.2: Conformal mapping of exponential and logarithm. … Magnify
2: 15.17 Mathematical Applications
§15.17(ii) Conformal Mappings
3: 16.23 Mathematical Applications
§16.23(iii) Conformal Mapping
The Bieberbach conjecture states that if n = 0 a n z n is a conformal map of the unit disk to any complex domain, then | a n | 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
§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 1 2 π , compare (4.28.8). …
6: 4.15 Graphics
§4.15(ii) Complex Arguments: Conformal Maps
Figure 4.15.7 illustrates the conformal mapping of the strip - 1 2 π < z < 1 2 π onto the whole w -plane cut along the real axis from - to - 1 and 1 to , where w = sin z and z = arcsin w (principal value). …
See accompanying text
Figure 4.15.7: Conformal mapping of sine and inverse sine. … Magnify
7: 23.20 Mathematical Applications
§23.20(i) Conformal Mappings
For examples of conformal mappings of the function ( z ) , 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
§22.18(ii) Conformal Mapping
With k [ 0 , 1 ] the mapping z w = sn ( z , k ) gives a conformal map of the closed rectangle [ - K , K ] × [ 0 , K ] onto the half-plane w 0 , with 0 , ± K , ± K + i K , i K mapping to 0 , ± 1 , ± k - 2 , 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
§29.18(iv) Other Applications
Triebel (1965) gives applications of Lamé functions to the theory of conformal mappings. …
10: Bibliography H
  • P. Henrici (1974) 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.
  • P. Henrici (1986) 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.