Sheet 06 - Exercise 6 PDF

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Exercise 6...

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Computational Photonics Group Department of Electrical and Computer Engineering Technical University of Munich

Partial Differential Equations in Electrical Engineering Exercise 6 Prof. Dr. Christian Jirauschek and Michael Riesch, M.Sc. Winter semester 2019/20

Conformal Mapping In this exercise we will use the conformal mapping formalism introduced in the lecture notes. Consider the angular region  o n

 Dφ ≡ z = |z|eiψ ∈ C 0 ≤ ψ ≤ φ .

Using the Green’s function for the half-space (upper half-plane)

G(w, wQ ) =

|w − wQ | 1 · ln , 2π |w − w ¯Q |

where w ¯Q is the complex conjugate, and the conformal mapping from Dφ to the upper half-plane π

w(z) = z φ , find the Green’s function for Dφ . Check your results for the quarter space (i.e., when φ = 2π)....