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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

[tex]\frac{f(z)}{z-z_{o}}[/tex] has a simple pole at [tex] z_{0} [/tex] with residue [tex]f(z_{o})[/tex] then the theorem says that if f(z) is analytic within C the value of f at some point [tex] z_{0} [/tex] within C is given by

[tex]f(z_{0})=\frac{1}{2\pi i} \oint_{C}\frac{f(z)}{z-z_{o}} dz[/tex]

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

thanxs

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# Cauchy theorem

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