Deformation, Path Indep. Thms for Ints. of Complex Functions

The Deformation and Path Independence Theorems for Integrals of Complex Functions

We will now state two very important theorems regarding integrals of complex functions.

The Deformation Theorem

Theorem 1 (The Deformation Theorem): Let $A \subseteq \mathbb{C}$ be open and let $f : A \to \mathbb{C}$ be analytic on $A$. If $\gamma_1$ and $\gamma_2$ are two piecewise smooth curves in $A$ that are homotopic to each other then $\displaystyle{\int_{\gamma_1} f(z) \: dz = \int_{\gamma_2} f(z) \: dz}$.

The deformation theorem is useful as it can reduce potentially difficult problems to simpler problems. For example, suppose that we want to compute the integral of a function $f$ along a complicated piecewise smooth curve $\gamma_1$. In other words, suppose that $\gamma_1$ is difficult in the sense that $\gamma_1$ may be difficult to parameterized or may result in attempting to evaluate a tedious integral. If $f$ is analytic on some open domain $A$ for which $\gamma_1$ is contained in, then if we can find a simpler piecewise smooth curve $\gamma_2$ homotopic to $\gamma_1$, then by the deformation theorem, we can evaluate the integral of $f$ along $\gamma_1$ as evaluating the integral of $f$ along $\gamma_2$.


The Path Independence Theorem

Theorem 2 (The Path Independence Theorem): Let $A \subseteq \mathbb{C}$ be open, simple (has no holes), and connected and let $f : A \to \mathbb{C}$ be analytic on $A$. If $\gamma_1$ and $\gamma_2$ are two piecewise smooth paths in $A$ from the points $z_1$ to the point $z_2$ ($z_1, z_2 \in A$) then $\displaystyle{\int_{\gamma_1} f(z) \: dz = \int_{\gamma_2} f(z) \: dz}$.
  • Proof: Let $A$ be an open, simple, and connected domain. If $\gamma_1$ and $\gamma_2$ are two piecewise smooth paths in $A$ from $z_1$ to $z_2$ then let $\gamma = \gamma_1 - \gamma_2$ . Then $\gamma$ is a closed, piecewise smooth curve in $A$. Since $A$ is simple and connected there are no holes in the interior of $\gamma$ so $\gamma$ is homotopic to a point in $A$. By Cauchy's integral theorem:
\begin{align} \quad 0 = \int_{\gamma} f(z) \: dz = \int_{\gamma_1 - \gamma_2} f(z) \: dz = \int_{\gamma_1} f(z) \: dz - \int_{\gamma_2} f(z) \: dz \end{align}
  • Thus:
\begin{align} \quad \int_{\gamma_1} f(z) \: dz = \int_{\gamma_2} f(z) \: dz \quad \blacksquare \end{align}

The path independence theorem is useful as it tells us that for open, simple, and connected domains $A$ for which $f$ is analytic that the integral of $f$ along any path from $z_1$ to $z_2$ is the same.

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