Taliyah Spencer

2022-05-03

Antiderivatives in complex analysis

Let U be an open set in $\mathbb{C}$ and $f:U\to E$ be holomorphic on U. Let now $\gamma $ be a path in U that begins with ${z}_{1}\in U$ and ends with ${z}_{2}\in U.$. I wonder for which assumptions the following formula ${\int}_{\gamma}\frac{df}{dz}(z)=f({z}_{2})-f({z}_{1})$ is true?

I have found many counterexamples (such as ${f}^{\prime}(z)=\frac{1}{z}$) 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 $\overline{U}$ and ${z}_{1}\in \mathrm{\partial}U$ is the above formula also true?

Let U be an open set in $\mathbb{C}$ and $f:U\to E$ be holomorphic on U. Let now $\gamma $ be a path in U that begins with ${z}_{1}\in U$ and ends with ${z}_{2}\in U.$. I wonder for which assumptions the following formula ${\int}_{\gamma}\frac{df}{dz}(z)=f({z}_{2})-f({z}_{1})$ is true?

I have found many counterexamples (such as ${f}^{\prime}(z)=\frac{1}{z}$) 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 $\overline{U}$ and ${z}_{1}\in \mathrm{\partial}U$ is the above formula also true?

Tyler Velasquez

Beginner2022-05-04Added 19 answers

Step 1

If f is holomorphic in $U\subset \mathbb{C}$ and $\gamma $ is a (piecewise differentiable) path in U joining ${z}_{1}$ and ${z}_{2}$ then

${\int}_{\gamma}{f}^{\prime}(z)\phantom{\rule{thinmathspace}{0ex}}dz={\int}_{0}^{1}{f}^{\prime}(\gamma (t)){\gamma}^{\prime}(t)\phantom{\rule{thinmathspace}{0ex}}dt={\textstyle [}f(\gamma (t)){{\textstyle ]}}_{t=0}^{t=1}=f({z}_{2})-f({z}_{1})$

Step 2

So that relation holds always, even in multiply connected domains.

${\int}_{|z|=1}\frac{1}{z}\phantom{\rule{thinmathspace}{0ex}}dz=2\pi i\ne 0$ is not a counter-example, because 1/z is not the derivative of a holomorphic function f in any domain containing the unit circle.

If f is holomorphic in $U\subset \mathbb{C}$ and $\gamma $ is a (piecewise differentiable) path in U joining ${z}_{1}$ and ${z}_{2}$ then

${\int}_{\gamma}{f}^{\prime}(z)\phantom{\rule{thinmathspace}{0ex}}dz={\int}_{0}^{1}{f}^{\prime}(\gamma (t)){\gamma}^{\prime}(t)\phantom{\rule{thinmathspace}{0ex}}dt={\textstyle [}f(\gamma (t)){{\textstyle ]}}_{t=0}^{t=1}=f({z}_{2})-f({z}_{1})$

Step 2

So that relation holds always, even in multiply connected domains.

${\int}_{|z|=1}\frac{1}{z}\phantom{\rule{thinmathspace}{0ex}}dz=2\pi i\ne 0$ is not a counter-example, because 1/z is not the derivative of a holomorphic function f in any domain containing the unit circle.

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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