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In complex analysis, a function φ: C → C, φ(x,y) = u(x,y) + iv(x,y) is analytic, or complex differentiable if its real and imaginary parts u and v satisfy the Cauchy-Riemann equations:

∂u ∂x = ∂v ∂y ∂v ∂x =− ∂u ∂y

Functions u and v for which these equations hold true are known as conjugate functions.