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Conjugate Functions

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 dv =(∂v/∂x)dx + (∂v/∂y)dy

Replacing ∂v/∂x by -∂u/∂y and ∂v/∂y bu ∂u/∂x, we get
dv =-(∂u/∂y)dx + (∂u/∂x)dy
v = -∫(∂u/∂y)dx + (∂u/∂x)dy
v = M dx + N dy


where M = -∂u/∂y and N = ∂u/∂x


v can be integrated as
M dx + N dy can be proved to be  an exact differential, thus v is determined.



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