Jan 18

Q: So, what's is complex analysis? Isn't it just function that take input a complex number, and output another complex number?

A: Well, not any such function is what we want to study, we care about super nice functions, called analytic functions.

Q: oh, I know what is an analytic function, these are just function that admits Taylor expansions at every points. And they all have names, like $\log(x), \sin(x), \exp(x)$.

A: hmm, partly right, analytic functions are those which admits Taylor series description locally at each point, but they may not all have names, they can be really wild, things that have infinitely many details which you can tweak about. In fact, we only require first order derivative to exists at all points.

Q: alright, then what are we going to learn about these functions? Why you say they are super nice?

A: Here are some cool facts, which we are going to learn and prove:

• local behavior determines global property
• regularity: even though we only require $f$ to admits first order derivative, it actually have all derivatives.
• contour integral: if we do line integral of the form $\int_C f(z) dz$, then we can wiggle the integration contour $C$ as long as $f$ remains nice on $C$.
• Special complex functions, Zeta function, Theta function, Gamma function.

OK, enough of that fancy stuff. Let's go back to the basics.

How to think about a complex number?

• real plus imaginary (good for addition)
• phase angle and radius (good for multiplication)
• exponential and log