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Milne Thomson Method

Method to find the conjugate function

Given: If f(z) = u + iv and u is known

To find: v the conjugate function.

Method.
We know that f'(z) =(∂u/∂x) + i(∂v/∂x),
 f'(z) =(∂u/∂x) - i(∂u/∂y) by C-R equations

If we write ∂u/∂x = φ1(x,y), ∂u/∂y = φ2(x,y)
f'(z) = φ1(x,y) - iφ2(x,y) or f'(z) = φ1(z,0) - iφ2(z,0)

On integrating,
f(z) = φ1(z,0) dz - iφ2(z,0) dz + c when u is given

Similarly
f(z) = ψ1(z,0) dz + iψ2(z,0) dz + c when v is given
∂v/∂y =ψ1(x,y), ∂v/∂x = ψ2(x,y)



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