Properties of Integrals of Complex Functions Along P.S. Curves

Properties of Integrals of Complex Functions Along Piecewise Smooth Curves

Recall from the Integrals of Complex Functions Along Piecewise Smooth Curves page that if $A \subseteq \mathbb{C}$ is open, $f : A \to \mathbb{C}$ is continuous, $\gamma : [a, b] \to \mathbb{C}$ is a piecewise smooth curve with $\gamma ([a, b]) \subset A$ where $a = a_0 < a_1 < ... < a_n = b$ is a partition on $[a, b]$] for which [[$\gamma'$ exists on each open subinterval $(a_k, a_{k+1})$ and is continuous on each closed subinterval $[a_k, a_{k+1}]$ for all $k \in \{0, 1, ..., n-1\}$, then the integral of $f$ along the curve $\gamma$ is:

(1)
\begin{align} \quad \int_{\gamma} f(z) \: dz = \int_a^b f(\gamma(t)) \cdot \gamma'(t) \: dt = \sum_{k=0}^{n-1} \int_{a_k}^{a_{k+1}} f(\gamma(t)) \cdot \gamma'(t) \: dt \end{align}

We will now state some basic properties of such integrals of complex functions.

 Theorem 1: Let $A \subseteq \mathbb{C}$ be open, and let $f, g : A \to \mathbb{C}$ be continuous functions. Let $\gamma : [a, b] \to \mathbb{C}$ be a piecewise smooth curve such that $\gamma ([a, b]) \subset A$. Then: a) $\displaystyle{\int_{\gamma} [f(z) + g(z)] \: dz = \int_{\gamma} f(z) \: dz + \int_{\gamma} g(z) \: dz}$ (Additivity property). b) $\displaystyle{\int_{\gamma} kf(z) \: dz = k \int_{\gamma} f(z) \: dz}$ for all $k \in \mathbb{C}$ (Homogeneity property).
• Proof of a) We have that:
(2)
\begin{align} \quad \int_{\gamma} [f(z) + g(z)] \: dz &= \int_a^b [f(\gamma(t)) + g(\gamma(t))] \cdot \gamma'(t) \: dt \\ &= \int_a^b [f(\gamma(t)) \cdot \gamma'(t) + g(\gamma(t)) \cdot \gamma'(t)] \: dt \\ &= \int_a^b f(\gamma(t)) \cdot \gamma'(t) \: dt + \int_a^b g(\gamma(t)) \cdot \gamma'(t) \: dt \\ &= \int_{\gamma} f(z) \: dz + \int_{\gamma} g(z) \: dz \quad \blacksquare \end{align}
• Proof of b) Let $k \in \mathbb{C}$. Then we have that:
(3)
\begin{align} \quad \int_{\gamma} kf(z) \: dz &= \int_a^b kf(\gamma(t)) \cdot \gamma'(t) \: dt \\ &= k \int_a^b f(\gamma(t)) \cdot \gamma'(t) \: dt \\ &= k \int_{\gamma} f(z) \: dz \quad \blacksquare \end{align}
 Theorem 2: Let $A \subseteq \mathbb{C}$ be open, and let $f, g : A \to \mathbb{C}$ be continuous functions. Let $\gamma : [a, b] \to \mathbb{C}$ be a piecewise smooth curve such that $\gamma ([a, b]) \subset A$. If $\overline{\gamma}$ is a reparameterization of $\gamma$ then $\displaystyle{\int_{\gamma} f(z) \: dz = \int_{\overline{\gamma}} f(z) \: dz}$.