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A holomorphic function is a function that is differentiable at every point on its domain. In order for it to be differentiable, it needs to satisfy the Cauchy-Riemann equation properties, such that:
f(z) = u(x,y) + iv(x,y)
ux = vy
vx = -uy
If that is so, then f'(z) = ux + ivx
Otherwise, if a function doesn't satisfy these conditions, we say that it's not holomorphic. For instance:
f(z) = z̅
Test with the following properties:
ux = vy
vx = -uy
z̅ is written as u(x,y) - iv(x,y). Take the partial derivatives of u(x,y) and v(x,y). Then:
ux = -vy
vx = -(-uy) = uy
Since the conditions don't hold, that function is not holomorphic.
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