Derivation of Poisson Integral Formula for the Half-Plane
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If is analytic in a domain containing the x-axis and the upper half-plane and in this domain, then the values of the harmonic function in the upper half-plane are given in terms of its values on the x-axis by
(y>0).
Below is an outline for the derivation, I just need to figure out how to justify the steps.
a) for the situation depicted above
b) for the same situation
c) Subtract these two equations to conclude that
Where Cr+ is the semicircular portion of ҐR
d) Show that the integral along Cr+ is bounded by
e) Let Rinfinity in the last equation and take the real part.
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If is analytic in a domain containing the x-axis and the upper half-plane and in this domain, then the values of the harmonic function in the upper half-plane are given in terms of its values on the x-axis by
(y>0).
Below is an outline for ...
Purchase this Solution
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