Complex Analysis: Holomorphic functions

[Potage11]

So my teacher explained what holomorphic functions were today. But it did not make much sense.
As I am attempting to do my homework, I realized that I still don't really know what a Holomorphic function is, let alone how to show that one is.

The questions looks like this:
show that f(z)=u(z)+iv(z) is holomorphic or not;
where u and v are given different values throughout the problem.

I was hoping someone could clarify what a holomorphic function is, and maybe show me a little trick as to how I should go about this problem.

Thanks

 

[SiddharthM]

holomorphic is another word for analytic - which means differentiable on some open set in the plane.

There is a difference between being analytic and being differentiable. For f to be analytic at a point z - it means that there is an open set containing x throughout which the function is differentiable. If you are differentiable ONLY at one point then you are NOT analytic.

f is differentiable at z iff the cauchy reimann equations are satisfied at that point. This is probably the easiest way to show a function is holomorphic.

 

[mathwonk]

a holomorphic function C-->C is a real differentiable function R^2-->R^2 whose derivative as a linear map is actually complex linear. this means the matrix of partials has the same entry in both diagonal entries and the off diagonal entries are negatives of each other.

i.e. to be holomorphic it suffices for a function u+iv to have continuous partials, which satisfy du/dx = dv/dy and du/dy = -dv/dx (Cauchy - Riemann equations).

e.g. u = ln(sqrt(x^2+y^2)), and v = arctan(y/x).i hope these work.