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 $\displaystyle{\int_{\gamma} f(z) \: dz = \int_{\overline{\gamma}} f(z) \: dz}$.