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Lecture24

# Lecture24 - Therefore the tangent mapping is a linear...

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Therefore, the tangent mapping is a linear fraction transformation, pre- ceded by an exponential mapping and followed by a clockwise rotation by 90 . With Z = e - 2 y e 2 ix , we see that the strip S := { z C | x ( - π/ 4 , π/ 4) } in the z -plane is mapped onto the right half of the Z -plane, X > 0. The linear fractional transformation maps the right half of the Z -plane onto the interior of the unit disk in the W -plane. A clockwise rotation by π/ 2 follows, so that w = tan z maps the strip S in the z -plane onto the interior of the unit disk in the w -plane. 18 Complex Analysis and Potential Theory The theory of solutions of Laplace’s equation (cf. Sec. 12.10) is called poten- tial theory. Laplace’s equations occurs in many fields such as gravitation, electrostatics, heat conduction, fluid flow etc. In two space dimensions, and in cartesian coordinates, we are interested in solutions of the second-order PDE ∆Φ = Φ xx + Φ yy = 0 , in Ω R 2 , (1011) for the (real-valued) potential Φ : Ω R . We know from Section 13.4 that solutions of (1011) with continuous second derivatives (harmonic functions) are closely related to holomorphic functions Φ + i Ψ where Ψ is a harmonic conjugate of Φ. The restriction to two space dimensions is often justified when the poten- tial Φ is independent of one of the space coordinates. In this last chapter of the lecture, we shall consider this connection and its consequences in detail and illustrate it by typical boundary value problems from electrostatics, heat conduction, and hydrodynamics. 18.1 Electrostatic Fields The electric force F [ N ] between charged particles is given by Coulomb’s law. This force is the gradient of a function Φ: F = q E , E = -∇ Φ, with the electric field E [Vm - 1 ] = [NC - 1 ] and the electrostatic potential Φ [V] (cf. Sec. 12.10; there we looked at Newton’s law of universal gravitation, but Coulomb’s law has a similar form). Equipotential surfaces are level sets of Φ const . ; lines in 2D). The electric field is perpendicular to these surfaces. 183

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Examples: 1. Two parallel plates kept at potentials Φ 1 , Φ 2 , respectively, extending to infinity. Choose a cartesian coordinate system ( x, y, z ) such that the plates are parallel to the yz -plane, and located at x = - 1 and x = 1. Due to translation invariance of the problem in both y - and z -direction, we may restrict our consideration to the x -direction, where we have a second-order ODE with Dirichlet boundary conditions: Φ ′′ = 0 , in ( - 1 , 1) , Φ( - 1) = Φ 1 , Φ(1) = Φ 2 . (1012) The general solution is given by Φ( x ) = ax + b , and the constants a, b are determined from the boundary conditions: Φ( - 1) = - a + b = Φ 1 Φ(1) = a + b = Φ 2 a = 1 2 2 - Φ 1 ) b = 1 2 1 + Φ 2 ) (1013) Therefore, the potential between two parallel plates is given by Φ( x ) = 1 2 2 - Φ 1 ) x + 1 2 1 + Φ 2 ) . (1014) The equipotential surfaces are parallel planes, x = const .
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