The F. T. of Calculus for Ints of Comp. Functs. Along P.S. Curves

Math Online's 2000th Page - 10:56 PM CST on February 25th, 2016

# The Fundamental Theorem of Calculus for 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 are now ready to state a very important theorem known as the Fundamental Theorem of Calculus for Curve Integrals. It is very similar to that of the normal Fundamental Theorem of Calculus presented in elementary calculus. It states that if $F$ is an analytic function on an open subset of $A$ of $\mathbb{C}$ and that $\gamma$ is a piecewise smooth curve in $A$ then the integral of $F$ along $\gamma$ can be obtained by taking the difference between $F$ evaluated at the terminal point of $\gamma$ and the initial point of $\gamma$.

 Theorem 1 (The Fundamental Theorem of Calculus for Curve Integrals): Let $A \subseteq \mathbb{C}$ be open, and let $F : A \to \mathbb{C}$ be a continuous function. Let $\gamma : [a, b] \to \mathbb{C}$ be a piecewise smooth curve such that $\gamma ([a, b]) \subset A$. If $F$ is analytic on $A$ then $\displaystyle{\int_{\gamma} F'(z) \: dz = F(\gamma(b)) - F(\gamma(a))}$.
• Proof:
(2)
\begin{align} \quad \int_{\gamma} F'(z) \: dz & = \int_{a}^{b} F'(\gamma(t)) \cdot \gamma'(t) \: dt \\ &= \int_a^b (F \circ \gamma)'(t) \: dt \quad (\mathrm{Chain \: rule}) \\ \end{align}
• Let $\sigma(t) = (F \circ \gamma)(t)$. Then $\sigma(t) = x(t) + iy(t)$, and $\sigma'(t) = (F \circ \gamma)'(t) = x'(t) + iy'(t)$ where $x'$ and $y'$ are continuous. So:
(3)
\begin{align} \quad \int_{\gamma} F'(z) \: dz &= \int_a^b x'(t) \: dt + i \int_a^b y'(t) \: dt \\ &= x(b) - x(a) + i[y(b) - y(a)] \quad (\mathrm{Fundamental \: Theorem \: of \: Calculus}) \\ &= [x(b) + iy(b)] - [x(a) + iy(a)] \\ &= \sigma(b) - \sigma(a) \\ &= (F \circ \gamma)(b) - (F \circ \gamma)(a) \\ &= F(\gamma(b)) - F(\gamma(a)) \quad \blacksquare \end{align}