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.

Katsutoshi Yamanoi has written:

'Holomorphic curves in Abelian varieties and intersections with higher codimensional subvarieties'

'On the truncated small function theorem in Nevanlinna theory'

A biholomorphism is a bijective holomorphism whose inverse is also holomorphic.

T. Suwa has written:

'Indices of vector fields and residues of singular holomorphic foliations' -- subject(s): Holomorphic functions, Index theory (Mathematics), Vector fields

David E. Barrett has written:

'Transverse symmetries and proper holomorphic maps'