I'm studying complex analysis on professor Tao's notes. I need to prove that the complex integral defined as such:
*Let $\gamma : [a,b] \to \mathbb{C}$ and $f: \gamma([a,b]) \to \mathbb{C}$ be complex functions. We define $\int_\gamma f$, if it exist, as the number such that for every $\epsilon \gt 0$ there exist $\delta$ such that, if $a = x_0 \lt x_1 \lt \cdots \lt x_n = b$ determines a partition of $[a,b]$ , $\max_{1 \le i \le n}| x_i - x_{i-1}| < \delta$ and $x_{i-1} \le \xi_i \le x_i$, then $\displaystyle \left|\sum_{i=1}^n f(\gamma(\xi_i))(\gamma(x_i) - \gamma(x_{i-1})) - \int_\gamma\right| \lt \epsilon$ *
is the same as the standard one, when $f$ is continous and $\gamma$ rectifiable and continously differentiable, in the sense that: $$ \int_\gamma f = \int_a^b\Re[f(\gamma(t))\gamma'(t)]\,dt + i \int_a^b\Im[f(\gamma(t))\gamma'(t)]\,dt$$
(it can be proven that $\int_\gamma f$ exists whenever $f$ is continous and $\gamma$ is rectifiable)
I tought I might show that the complex riemann sums of the left side are arbitrarily close to real ones of the left side, then use the triangular inequality to show the equality above.
If $a = x_0 \lt x_1 \lt \cdots \lt x_n = b$ determines a partition of $[a,b]$, with $1 \le \xi_{i-1} \le x_i$ for $1 \le i \le n$ and $\max_{1 \le i \le n}| x_i - x_{i-1}| < \delta$ for an appropriate $\delta$, we have $$ \left| \sum_{i=1}^n f(\gamma(\xi_i))(\gamma(x_i) - \gamma(x_{i-1})) - \sum_{i=1}^n \Re[f(\gamma(\xi_i))\gamma(\xi_i)](x_i - x_{i-1}) - i\sum_{i=1}^n \Im[f(\gamma(\xi_i))\gamma(\xi_i)](x_i - x_{i-1}) \right| = $$
$$ \left| \sum_{i=1}^n f(\gamma(\xi_i))(\gamma(x_i) - \gamma(x_{i-1})) - \sum_{i=1}^n \Re[f(\gamma(\xi_i))\gamma(\xi_i)](x_i - x_{i-1}) - i\Im[f(\gamma(\xi_i))\gamma(\xi_i)](x_i - x_{i-1}) \right| = $$
$$ \left|\sum_{i=1}^n f(\gamma(\xi_i))(\gamma(x_i) - \gamma(x_{i-1})) - \sum_{i=1}^n f(\gamma(\xi_i))\gamma(\xi_i)](x_i - x_{i-1})\right| = $$
$$ \left|\sum_{i=1}^n f(\gamma(\xi_i))[(\gamma(x_i) - \gamma(x_{i-1})) - \gamma(\xi_i)(x_i - x_{i-1})]\right| $$
But then i don't know what to do, since there does not exist a mean value theorem for complex numbers. How can i proceed? Or should I drop that and try a different method?
There is no mean value theorem, but since $\gamma'$ is continuous you have $$ \gamma'(\xi)-\gamma'(x_i)=\text{Error} $$ with $Error\to 0$ as the partition is refined. This error does not contribute in the limit you have because $Error\times (x_i-x_{i-1})$ goes to zero faster than the diameter of the partition. Then, you can safely replace $\gamma'(\xi)$ by $\gamma'(x_{i-1})$ in your last formula and, by the Peano form of the remainder in Taylor's expansion, the expression in brackets also goes to zero faster than the diameter of the partition, thus giving zero in the limit for the sum. I can give more details if you need them.