TITLE: Regarding Theorem 1.9, Conway, complex Analysis
QUESTION [0 upvotes]: I have a doubt in equation (1.11) in Chapter 4, Section 1 of J.B Conway’s, functions of one complex variable.
I know that the estimation of the integral in equation 1.10, comes from the previous theorem 1.4.
But what about the integral  $\int_{a}^{b} f(t) \gamma '(t) dt$? How is it approximated? By Riemann sums?

REPLY [0 votes]: Since $f\gamma'$ is Riemann integrable, then, for every $\varepsilon>0$, there is some $\delta>0$ such that, for every partition $P$ of $[a,b]$ whe $|P|<\delta$, we have$$\left|\overline{\sum}(f\gamma',P)-\underline{\sum}(f\gamma',P)\right|<\frac\varepsilon2.$$But if $P=\{t_0(=a),t_1,\ldots,t_n(=b)\}$, with $t_0<t_1<\cdots<t_n$, and if, for each $k\in\{1,2,\ldots,n\}$, $\tau_k\in[t_{k-1},t_k]$, we have$$\sum_{k=1}^nf(\tau_k)\gamma'(\tau_k)(t_k-t_{k-1})\in\left[\underline{\sum}(f\gamma',P),\overline{\sum}(f\gamma',P)\right].\tag1$$but $\int_a^bf(t)\gamma'(t)\,\mathrm dt$ also belongs to the RHS of $(1)$. Since the length of that interval is smaller than $\varepsilon2$,$$\left|\int_a^bf(t)\gamma'(t)\,\mathrm dt-\sum_{k=1}^nf(\tau_k)\gamma'(\tau_k)(t_k-t_{k-1})\right|<\frac\varepsilon2.$$
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