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:
(1)
\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:
(2)
\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.