Applications for holomorphic functions? Could anyone give me an insight into practical applications of holomorphic functions (I am using the term in the way in which it is related to Riemann's work)? 

A very interesting consequence of holomorphic functions is that as they satisfy the Cauchy-Riemann equations, they automatically satisfy Laplace's equation. Because of this, we can model 2-D potential fluid flow based on holomorphic functions. For example fluid flowing past a cylinder of radius $a$ can be imagined by taking constant values of the imaginary part of the complex function $$\Omega(z) = U\left(z+\frac{a^2}{z}\right)$$ resulting in this fluid flow:

where the streamline in blue visualises the dividing streamline where stagnation points occur on the solid body. 
