Harmonic Functions and Conformal Mapping

2010 ◽  
pp. 139-198
Keyword(s):  
Conformal Mapping ◽  
Harmonic Functions

The Robin function and conformal welding – A new proof of the existence

2020 ◽  
Vol 27 (1) ◽  
pp. 43-51
Author(s):  
Bodo Dittmar
Keyword(s):  
Conformal Mapping ◽  
Boundary Value Problem ◽  
Harmonic Functions ◽  
Mixed Boundary ◽  
Mixed Boundary Value Problem ◽  
Uniformization Theorem ◽  
Robin Function ◽  
Connected Case ◽  
Simply Connected ◽  
Mathematical Physicist

AbstractGreen’s function of the mixed boundary value problem for harmonic functions is sometimes named the Robin function {R(z,\zeta\/)} after the French mathematical physicist Gustave Robin (1855–1897). The aim of this paper is to provide a new proof of the existence of the Robin function for planar n-fold connected domains using a special version of the well-known Koebe’s uniformization theorem and a conformal mapping which is closely related to the Robin function in the simply connected case.

scholarly journals Discrete Harmonic Functions in the Three-Quarter Plane

2021 ◽  
Author(s):  
Amélie Trotignon
Keyword(s):  
Functional Equation ◽  
Conformal Mapping ◽  
Boundary Value Problem ◽  
Random Walks ◽  
Harmonic Functions ◽  
Symmetric Convex ◽  
Planar Lattice ◽  
Quarter Plane ◽  
Lattice Random Walks ◽  
Discrete Harmonic Functions

AbstractIn this article we are interested in finding positive discrete harmonic functions with Dirichlet conditions in three quadrants. Whereas planar lattice (random) walks in the quadrant have been well studied, the case of walks avoiding a quadrant has been developed lately. We extend the method in the quarter plane—resolution of a functional equation via boundary value problem using a conformal mapping—to the three-quarter plane applying the strategy of splitting the domain into two symmetric convex cones. We obtain a simple explicit expression for the algebraic generating function of harmonic functions associated to random walks avoiding a quadrant.

A novel analytical calculation of magnetic field in the slotted air gap of spoke-type permanent-magnet machines using conformal mapping

2020 ◽  
pp. 1-15
Author(s):  
Jianqi Li ◽  
Yu Zhou ◽  
Jianying Li
Keyword(s):  
Magnetic Field ◽  
Conformal Mapping ◽  
Permanent Magnet ◽  
Flux Density ◽  
Analytical Method ◽  
Electromotive Force ◽  
Analytical Calculation ◽  
Air Gap ◽  
Permanent Magnet Machines ◽  
Peak Value

This paper presented a novel analytical method for calculating magnetic field in the slotted air gap of spoke-type permanent-magnet machines using conformal mapping. Firstly, flux density without slots and complex relative air-gap permeance of slotted air gap are derived from conformal transformation separately. Secondly, they are combined in order to obtain normalized flux density taking account into the slots effect. The finite element (FE) results confirmed the validity of the analytical method for predicting magnetic field and back electromotive force (BEMF) in the slotted air gap of spoke-type permanent-magnet machines. In comparison with FE result, the analytical solution yields higher peak value of cogging torque.

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