yopopolin10d

2022-05-01

Antiderivative simply connected region

Why do analytic functions always have an antiderivative on a simply connected region?

Why do analytic functions always have an antiderivative on a simply connected region?

kubistiedt

Beginner2022-05-02Added 17 answers

Step 1

Cauchy's Theorem tells us that $\int}_{{\gamma}_{1}}f(x)dz={\displaystyle {\int}_{{\gamma}_{2}}f(x)dz$ whenever ${\gamma}_{1}{\gamma}_{2}$ are homotopic, simple curves sharing the same endpoints in f's domain of analyticity $\mathrm{\Omega}.$. We can use this fact to define the antiderivative of f in $\mathrm{\Omega}$ as follows: $F(\omega )={\displaystyle {\int}_{\gamma}f(z)dz,}$, where $\gamma $ is a contour connecting a fixed point ${z}_{0}\in \mathrm{\Omega}$ to $\omega .$.

Step 2

Analyticity is required simply so that F is well-defined (in general it is not.) Using a parametrization $\rho (t)={\omega}_{0}+(\omega -{\omega}_{0})t,0\le t\le 1,$, you can show that the difference quotient satisfies $\frac{F(\omega )-F({\omega}_{0})}{\omega -{\omega}_{0}}={\displaystyle {\int}_{0}^{1}f({\omega}_{0}+t(\omega -{\omega}_{0}))dt.}$.

Cauchy's Theorem tells us that $\int}_{{\gamma}_{1}}f(x)dz={\displaystyle {\int}_{{\gamma}_{2}}f(x)dz$ whenever ${\gamma}_{1}{\gamma}_{2}$ are homotopic, simple curves sharing the same endpoints in f's domain of analyticity $\mathrm{\Omega}.$. We can use this fact to define the antiderivative of f in $\mathrm{\Omega}$ as follows: $F(\omega )={\displaystyle {\int}_{\gamma}f(z)dz,}$, where $\gamma $ is a contour connecting a fixed point ${z}_{0}\in \mathrm{\Omega}$ to $\omega .$.

Step 2

Analyticity is required simply so that F is well-defined (in general it is not.) Using a parametrization $\rho (t)={\omega}_{0}+(\omega -{\omega}_{0})t,0\le t\le 1,$, you can show that the difference quotient satisfies $\frac{F(\omega )-F({\omega}_{0})}{\omega -{\omega}_{0}}={\displaystyle {\int}_{0}^{1}f({\omega}_{0}+t(\omega -{\omega}_{0}))dt.}$.

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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