Conformal Slit Maps in Applied Mathematics

Abstract

Conformal slit maps play a fundamental theoretical role in analytic function theory and potential theory. A lesser-known fact is that they also have a key role to play in applied mathematics. This review article discusses several canonical conformal slit maps for multiply connected domains and gives explicit formulae for them in terms of a classical special function known as the Schottky-Klein prime function associated with a circular preimage domain. It is shown, by a series of examples, that these slit mapping functions can be used as basic building blocks to construct more complicated functions relevant to a variety of applied mathematical problems.2010 Mathematics subject classification: primary 30C20; secondary 31A15.

Secondary Schottky–Klein prime functions associated with multiply connected planar domains

“…The primary S-K prime function on the Schottky double of a planar domain has already been shown in recent years to afford numerous advantages, not to mention key insights, into solving problems in multiply connected domains [1][2][3]. We expect the secondary S-K prime functions explored here to have many similar applications.…”

“…Problems involving multiply connected domains are ubiquitous in applications, and the past decade has seen a resurgence of interest in mathematical methods for their solution. A promising approach, reviewed in its various guises in references [1][2][3], makes use of the function theory on a certain compact Riemann surface naturally associated with any multiply connected planar domain. This Riemann surface is known as the Schottky double of the domain.…”

Solving Wiener–Hopf problems without kernel factorization

“…It is a future challenge to understand how the methods given herein might be adapted to such generalized situations. This paper has been inspired by the authors' interests in solving mixed boundary value problems for Stokes flows motivated by technological challenges arising in microfluidics and other low Reynolds number flow situations [14][15][16][17]. In this context, several problems arise where it is not clear how one might even attempt to apply traditional Wiener-Hopf methods; the approach of this paper, on the other hand, can be directly extended to these cases.…”

“…are some coefficients. The reader is referred to [14][15][16][17] for further discussion of closely related problems of no-slip/no-shear boundary value problems in Stokes flows. The flow is antisymmetric about y = 0; this implies that the Goursat functions satisfy the following conditionsf…”

Numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and straight slit regions

“…[13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. Various applications of conformal mappings in science and engineering are considered in, e.g., [22,[32][33][34][35].…”