- #1
mathsciguy
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I was reading about Cauchy-Goursat theorem and one step in the proof stumped me. It's the easier one, that is, Cauchy's proof that requires the complex valued function f be analytic in R, and f' to be continuous throughout the region R interior to and on some simple closed contour C. So that the contour integral around C is equal to zero.
Also let f(z) = u(x,y) + i v(x,y).
The proof used the hypothesis of Green's theorem which required the two real functions u and v and their first order partial derivatives on R to be continuous. I was thinking that if f is already analytic in R, wouldn't that already require u and v to be continuous in R and its first order partial derivatives? Since the condition for the differentiability of f requires it to be so? Doesn't that mean that the assumption that f' be continuous unnecessary?
Edit: Oops, ok, I think I got it. I have confused myself with the theorem for 'sufficient condition for differentiability', which says that u and have to be continuous (and they have to obey the Cauchy-Riemann equations) in R. If such conditions are met by f, it implies its differentiability, but is the converse true?
I might have skipped a lot of stuff about the theorem and the proof so my question might be confusing, but hopefully someone well acquainted with the topic would step in and help.
Also let f(z) = u(x,y) + i v(x,y).
The proof used the hypothesis of Green's theorem which required the two real functions u and v and their first order partial derivatives on R to be continuous. I was thinking that if f is already analytic in R, wouldn't that already require u and v to be continuous in R and its first order partial derivatives? Since the condition for the differentiability of f requires it to be so? Doesn't that mean that the assumption that f' be continuous unnecessary?
Edit: Oops, ok, I think I got it. I have confused myself with the theorem for 'sufficient condition for differentiability', which says that u and have to be continuous (and they have to obey the Cauchy-Riemann equations) in R. If such conditions are met by f, it implies its differentiability, but is the converse true?
I might have skipped a lot of stuff about the theorem and the proof so my question might be confusing, but hopefully someone well acquainted with the topic would step in and help.
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