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4-a-define-f-r-gt-r-by-f-z-z-2-and-let-v-w-r-2-gt-r-be-the-real-and-imaginary-parts-of-f-pa860

4: (a) Define

`f:R->R`

by

`f(z)=z^(2)`

and let

`v,w:R^(2)->R`

be the real and imaginary parts of

`f`

so that

`f(x+iy)=v(x,y)+iw(x,y)`

. Show that

`v`

and

`w`

both satisfy Laplace's equation. (b) Solve Dirichlet's problem for

`u=u(x,y)`

on the square

`0<=x<=1,0<=y<=1`

satisfying the boundary conditions

`u(x,0)=x,u(x,1)=x-1,u(0,y)=-y^(2)`

and

`u(1,y)=1-y^(2)`

. Hint: use

`v(x,y)`

from Part (a) and notice that

`u(x,1)!=v(x,1)`

.