Definitions, Theorems, Examples, Topics for Asmar 2nd edition
Chapter 7: 7.5 only

 7.5 The Poisson Integral Formula
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  u_xx + u_yy = 0 on |x|<inf, y>0,
  u(x,0)=f(x)

 POISSON KERNEL
   P[y](x) = sqrt(2/Pi) y/(x^2+y^2)
   LEMMA. FT[P[y]] = exp(-y|w|)
   THEOREM 1. Laplace's PDE with boundary condition u(x,0)=f(x) has
              solution
                u(x,y)=(P[y]*f)(x)
                      = (y/Pi)times integral of f(s)/((x-s)^2+y^2)
                          for s=-inf to s=inf.
   DERIVATION. Start with h(w)=FT[u(x,y)] for fixed y. Transform the PDE 
   and the boundary condition to get (iw)^2h(w) + h''(w) = 0, h(0)=FT[f].
   Solving the ODE under the assumption that h is bounded gives 
   h = A exp(-wy) + B exp(yw) with A=0 for w<0 and B=0 for w>0 (y is 
   always positive). Then h(w)=F(w)exp(-y|w|)=F(w)FT[P[y]]=FT[P[y]*f], the
   last equality by the convolution theorem. Then h(w)=FT[u(x,y)] implies 
   by cancelation that w(x,y)=(P[y]*f)(x).

 EXAMPLE 1. Solve the Poisson problem for f(x)=100 pulse(x,-1,1).
            Find an equation for the isotherms.
 ANSWER: u(x,y)=(100/Pi)(arctan(v1)+arctan(v2)) where 
         v1=(1+x)/y and v2= (1-x)/y.
         The isotherms are x^2 + y^2 - 2cy = 1, where c=cot(T Pi/100).