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w=u(x+iy)*(x^2 - y^2 +k^2 + i(2xy))^-.5

We let N=x^2-y^2+k^2

M=2xy

R^2=(N^2+M^2)^2

theta=tan^-1(M/N)

using this, now

w=u(x+iy)*(cos(theta/2)-isin(theta/2))*(x^2 - y^2 +k^2 )^2 + (2xy)^2 )^-.25

I don't get that part. Btw, it simplifies to

w=u(x+iy)*(cos(theta/2)-isin(theta/2)*(R^2)^-.25