Is there a way to determine immediately if a function is holomorphic or not? I started complex analysis a few weeks ago and we are at the part where we are introduced to the Laurent expansions! I noticed particularly that in some of the solved exercise examples that it is immediately mentioned that "this function is holomorphic on" a certain set! I know that to prove that a function is holomorphic i guess you can use the Caushy-Goursat(dont remember if this was the name of the formula. Im talking about the one where you compare the partial derivatives) theorem but is there lets say some tricks you can use to determine if a function is holomorphic or not? In all honesty i still dont understand what it means for a complex function to be holomorphic? What does it mean and how does it help us? Can someone help me?
 A: You probably mean the Cauchy-Riemann equations. This is a rather unspecific question, but here we go, at least giving some general directions. A complex function is holomorphic on a domain if it is complex differentiable in that domain. A function $f$ is is complex differentiable at a point $z$ if the complex limit
$$\lim_{h \to 0} \frac{f(z+h)-f(z)}{h}$$
exists.
One can show, and this is one of the main features of a typical introductory course in complex analysis, that holomorphicity is equivalent to analyticity. A function is analytic if it is locally equal to its Taylor expansion.
You can immediately see that a function is holomorphic if it built up from holomorphic functions by addition, subtraction, multiplication, division and composition. The elementary functions polynomials, the exponential function and trigonometric function are all holomorphic. Rational functions are also holomorphic on all domains where its denominator is non-zero. Logarithmic functions are holomorphic except at its branch cut.
I also add that a nice and applied introduction to complex analysis introducing these topics in an easily accessible manner is Saff & Snider: Fundamentals of Complex Analysis.
