Method to find the conjugate function
Given: If f(z) = u + iv and u is knownTo find: v the conjugate function.Method. We know that dv = (∂v/∂x)dx + (∂v/∂y)dyReplacing ∂v/∂x by -∂u/∂y and ∂v/∂y bu ∂u/∂x, we get dv = -(∂u/∂y)dx + (∂u/∂x)dyv = -∫(∂u/∂y)dx + ∫(∂u/∂x)dy v = ∫M dx + ∫N dy where M = -∂u/∂y and N = ∂u/∂x v can be integrated as can be proved to be an exact differential, thus v is determined.
∫M dx + ∫N dy |

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