Antiderivatives in complex analysis Let U be an open set in <mrow class="MJX-TeXAtom-ORD"> <

Taliyah Spencer

Taliyah Spencer

Answered question

2022-05-03

Antiderivatives in complex analysis
Let U be an open set in C and f : U E be holomorphic on U. Let now γ be a path in U that begins with z 1 U and ends with z 2 U .. I wonder for which assumptions the following formula γ d f d z ( z ) = f ( z 2 ) f ( z 1 ) is true?
I have found many counterexamples (such as f ( z ) = 1 z ) where the above formula fails if the set U is not simply connected. Here is a related result.
Another question, if f is well defined on U ¯ and z 1 U is the above formula also true?

Answer & Explanation

Tyler Velasquez

Tyler Velasquez

Beginner2022-05-04Added 19 answers

Step 1
If f is holomorphic in U C and γ is a (piecewise differentiable) path in U joining z 1 and z 2 then
γ f ( z ) d z = 0 1 f ( γ ( t ) ) γ ( t ) d t = [ f ( γ ( t ) ) ] t = 0 t = 1 = f ( z 2 ) f ( z 1 )
Step 2
So that relation holds always, even in multiply connected domains.
| z | = 1 1 z d z = 2 π i 0 is not a counter-example, because 1/z is not the derivative of a holomorphic function f in any domain containing the unit circle.

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