MHB 1st Derivative of Cauchy Integral formula

ognik
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Hi - I know the final result for the n'th derivative, I am looking though at getting an expression for the 1st derivative of f(z).

From $ f({z}_{0}) = \frac{1}{2\pi i} \oint_{c} \frac{f(z)}{z - {z}_{0}}dz $ we get

$ \frac{f({z}_{0} + \delta {z}_{0}) -{f({z}_{0}}) }{\delta {z}_{0}} =
\frac{1}{2\pi i \delta {z}_{0} } (\oint_{c} \frac{f(z)}{z - {z}_{0} - \delta {z}_{0} } dz - \oint_{c} \frac{f({z}_{0})}{z - {z}_{0}}dz ) $

Where does the 2nd integral on the right come from?
 
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ognik said:
Hi - I know the final result for the n'th derivative, I am looking though at getting an expression for the 1st derivative of f(z).

From $ f({z}_{0}) = \frac{1}{2\pi i} \oint_{c} \frac{f(z)}{z - {z}_{0}}dz $ we get

$ \frac{f({z}_{0} + \delta {z}_{0}) -{f({z}_{0}}) }{\delta {z}_{0}} =
\frac{1}{2\pi i \delta {z}_{0} } (\oint_{c} \frac{f(z)}{z - {z}_{0} - \delta {z}_{0} } dz - \oint_{c} \frac{f({z}_{0})}{z - {z}_{0}}dz ) $

Where does the 2nd integral on the right come from?

I think the 2nd integral on the right should just be a copy of your original expression, right? That is, you should have
$$ \frac{f({z}_{0} + \delta {z}_{0}) -{f({z}_{0}}) }{\delta {z}_{0}} =
\frac{1}{2\pi i \delta {z}_{0} } \left(\oint_{c} \frac{f(z)}{z - {z}_{0} - \delta {z}_{0} } dz - \oint_{c} \frac{f({z})}{z - {z}_{0}}dz \right). $$
 
It's not quite the same, it has $ f({z}_{o}) $ on top instead of $ f({z}) $ and also the $\delta {z}_{o} $ in the divisor in front.

Saying that told me what I was missing (and also what you were saying), it is of course the $ - f({z}_{o}) $ part of the expression. Thanks Ackbach, got it now.
 
I posted this question on math-stackexchange but apparently I asked something stupid and I was downvoted. I still don't have an answer to my question so I hope someone in here can help me or at least explain me why I am asking something stupid. I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem: Let ##f:D\to\mathbb{C}## be an anlytic function over a simply connected region ##D##. If ##a## and ##z## are part of...
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