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Laplacian, real/imaginary parts, derivatives of complex numbers

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Studying for a midterm and could not get the following practice problems.

1. Find the Laplacian (d^2f/dx^2 + d^2f/d^y^2) for
a. f(x,y)=x^2-y^2
b. f(x,y)= (x^2-y^2) / (x^2+y^2) ((x,y) not equal to (0,0))
c. f(x,y)=(x^2-y^2) / (x^2+y^2)^2 ((x,y) not equal to (0,0))

2. Calculate the real part and the imaginary part of the complex polynomial p(z)= z^3-3z. Do the same for q(z)=(|z|^2)z + zbar. (here, z= x +iy, x,y belgon to R)

3. Find df/dz for p(z) and q(z) above, e^(sqrt(x^2+y^2))(cos (x)+ i sin (y)), and xyz. (here, d/dz=1/2(d/dx - i d-dy))

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This solution helps with various complex analysis problems.

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