Thread: Fundamental Theorem of Complex Calculus

1. Fundamental Theorem of Complex Calculus

Hello everyone, I was having trouble with the proof of this and I was wondering if anyone could help me!

Let $f:S \rightarrow \mathbb{C}$ be analytic (where S is an open set), and such that if $\gamma : [a,b] \rightarrow S$ is a piecewise continuously differentiable path in S, then if f' is continuous on $\gamma^*$(which is always true anyway), and $z_1:= \gamma (a), z_2:= \gamma (b),$

then $\int_{\gamma} f'(z)dz = f(z_2)-f(z_1)$

PROOF: Write $\gamma$as $\gamma_1 + \gamma_2 + \ldots + \gamma_n$, where each $\gamma_i : [t_{i-1}, t_i] \rightarrow S$ is a simple smooth path, with $\gamma_i (t_i) = \gamma_{i+1} (t_i), \gamma_1 (t_1) := \gamma(a), \gamma_n (t_n) := \gamma (b).$

Then $\int_{\gamma_i} f'(z) dz = \int_{t_{i-1}}^{t_i} f '(\gamma_i (t) ) \gamma_i ' (t) dt = \int_{t_{i-1}}^{t_i} \dfrac{d}{dt} f ( \gamma_i (t) ) dt =$ now here I have a problem because it says this line is equal to

$
\left[ f(\gamma_i (t)) \right]_{t_{i-1}}^{t^i}
$
by the fundamental theorem of calculus for REAL functions. however $f \circ \gamma_i$ is not real valued is it? so why can i use it here?

2. Just apply the fundamental theorem of calculus to the real and imaginary parts of the function. Indeed, $\int \phi(t)dt=\int {\rm Re}(\phi(t))dt+i\int {\rm Im}(\phi(t))dt$ for $\phi:\mathbb{R}\to\mathbb{C}$ (and a function is differentiable iff both its real and imaginary parts are). Similarly, most theorems like integration by part or change of variables adapt seamlessly to complex-valued functions.

3. ah I see this now! thanks very much