What is the Significance of the Cauchy Integral Theorem in Complex Analysis?

[kexue]

Could someone tell me what there is so astonishing about the Cauchy integral theorem? No that I doubt that it is, I simply and obviously do not understand it fully. My main issue is that a closed real line integral naturally gives zero and so no big deal that what happens in the complex case. So what the big fuss about?

thank you

 

[elibj123]

I'm afraid, there are many line integral which do not give zero.

Try the field:

\vec{F}(x,y)=\frac{x}{x^{2}+y^{2}}\hat{i}+\frac{-y}{x^{2}+y^{2}}\hat{j}

Around the unit circle.

(If you study complex integration you will find this integral quite familiar)

Also, Cauchy Integral Theorem holds of course only for an analytic function. Using this theorem, you can derive Cauchy Integral Formula, which is too quite "astonishing" and from there you've many conclusions derived about analytic functions, that reveal how powerful they are.

 

[wofsy]

The Cauchy Integral Theorem says that for an analytic function in a domain, its value at any point in the interior of a domain can be determined from its values on the boundary of the domain. Powerful stuff.

 

[Count Iblis]

Actually, we should say "holomorphic" (i.e. complex differentiable) not "analytic". The fact that holomorphic functions are analytic is implied precisely by Cauchy's Integral Theorem. I think if you only look at analytic functions from the start, you could give a far simple proof of Cauchy's theorem. But the nontrivial result of complex analysis is that a function that is only assumed to be differentiable is analytic.

 

[n!kofeyn]

Actually, we should say "holomorphic" (i.e. complex differentiable) not "analytic".

That's my main gripe about complex analysis books. They use analytic for complex differentiable, and then you find out later that it's really called holomorphic since analytic should be reserved to mean that a function is equal to it's Taylor series in a small neighborhood. I see this issue and confusion constantly arise in these threads.