What is holomorphic function?

Answer 1

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.