On a property of kernel functions of the Schwarz formula for a finitely connected circular domain

1972 ◽  
Vol 23 (3) ◽  
pp. 329-331 ◽  
Author(s):  
L. E. Dunduchenko
Keyword(s):  
Circular Domain ◽  
Kernel Functions ◽  
Schwarz Formula ◽  
Connected Circular Domain

The Schwarz–Christoffel mapping to bounded multiply connected polygonal domains

2005 ◽  
Vol 461 (2061) ◽  
pp. 2653-2678 ◽  
Author(s):  
Darren Crowdy
Keyword(s):  
Mapping Function ◽  
Circular Domain ◽  
Schottky Group ◽  
Polygonal Domain ◽  
Schottky Groups ◽  
Image Domain ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Connected Circular Domain ◽  
Natural Way

A formula for the generalized Schwarz–Christoffel mapping from a bounded multiply connected circular domain to a bounded multiply connected polygonal domain is derived. The theory of classical Schottky groups is employed. The formula for the derivative of the mapping function contains a product of powers of Schottky–Klein prime functions associated with a Schottky group relevant to the circular pre-image domain. The formula generalizes, in a natural way, the known mapping formulae for simply and doubly connected polygonal domains.

scholarly journals Periodic Schwarz–Christoffel mappings with multiple boundaries per period

2019 ◽  
Vol 475 (2228) ◽  
pp. 20190225 ◽  
Author(s):  
Peter J. Baddoo ◽  
Darren G. Crowdy
Keyword(s):  
Circular Domain ◽  
Target Domain ◽  
Multiply Connected ◽  
Single Period ◽  
Straight Line ◽  
Finite Set ◽  
Periodic Configuration ◽  
Parameter Problem ◽  
Prime Function ◽  
Connected Circular Domain

We present an extension to the theory of Schwarz–Christoffel (S–C) mappings by permitting the target domain to be a single period window of a periodic configuration having multiple polygonal (straight-line) boundaries per period. Taking the arrangements to be periodic in the x -direction in an ( x ,  y )-plane, three cases are considered; these differ in whether the period window extends off to infinity as y  →  ± ∞, or extends off to infinity in only one direction ( y  →  + ∞ or y  →  − ∞), or is bounded. The preimage domain is taken to be a multiply connected circular domain. The new S–C mapping formulae are shown to be expressible in terms of the Schottky–Klein prime function associated with the circular preimage domains. As usual for an S–C map, the formulae are explicit but depend on a finite set of accessory parameters. The solution of this parameter problem is discussed in detail, and illustrative examples are presented to highlight the essentially constructive nature of the results.

Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions

2007 ◽  
Vol 142 (2) ◽  
pp. 319-339 ◽  
Author(s):  
DARREN CROWDY
Keyword(s):  
Real Axis ◽  
Mapping Function ◽  
Circular Domain ◽  
Connected Region ◽  
Polygonal Domain ◽  
The Real ◽  
Multiply Connected ◽  
Region Exterior ◽  
New Formula ◽  
Connected Circular Domain

AbstractA formula for the generalized Schwarz–Christoffel conformal mapping from a bounded multiply connected circular domain to an unbounded multiply connected polygonal domain is derived. The formula for the derivative of the mapping function is shown to contain a product of powers of Schottky–Klein prime functions associated with the circular preimage domain. Two analytical checks of the new formula are given. First, it is compared with a known formula in the doubly connected case. Second, a new slit mapping formula from a circular domain to the triply connected region exterior to three slits on the real axis is derived using separate arguments. The derivative of this independently-derived slit mapping formula is shown to correspond to a degenerate case of the new Schwarz–Christoffel mapping. The example of the mapping to the triply connected region exterior to three rectangles centred on the real axis is considered in detail.

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