Exploring Cauchy's Theorem: A Journey into Complex Analysis

[steven187]

hello all

well i am going to slowly research my way into complex analysis and I decided to start with cauchys theorem i hope this is the best part to start with, well anyway it says that if f(z) is analytic and

$$\frac{f(z)}{z-z_{o}}$$ has a simple pole at $$z_{0}$$ with residue $$f(z_{o})$$ then the theorem says that if f(z) is analytic within C the value of f at some point $$z_{0}$$ within C is given by
$$f(z_{0})=\frac{1}{2\pi i} \oint_{C}\frac{f(z)}{z-z_{o}} dz$$

it would muchly be appreciated if someone could give me an in depth explanation of what this is saying especially that weird looking integral sign and the terms pole, analytic, residue and an example of how it is used would be helpful

thanxs

 

[shmoe]

Leaping into Cauchy's theorem is not going slowly. Have you tried looking up all the terms you don't understand in whatever references you have?

 

[steven187]

hello
wow i must of chose something that is at the far end of complex analysis well i looked up the terms the only one could slightly understand is analytic functions which from my understanding is that if you take the taylor series of a function and evaluate the remainder term as n goes to infinity, if it equals zero then it is analytic? what would be a graphical explanation of a function being analytic? by the way where would be a good place to start with complex analysis?

steven

 

[shmoe]

The beginning is a good place to start. Pick up a text and go. Really you can't skip the fundamental things and expect to understand the more advanced stuff.

 

[steven187]

will do thanxs for the advice shmoe

steven