Given a curve $\gamma: [a, b] \mapsto \mathbb{C}$, how do we find it's length ?

$$\begin{aligned} \text{length}(\gamma) &\approx \sum_{j=0}^n \,\left|\, \gamma(t_{j+1}) - \gamma(t_j) \,\right|\, \\ &\approx \sum_{j=0}^n \frac{\,\left|\, \gamma(t_{j+1}) - \gamma(t_j) \,\right|\,}{t_{j+1} - t_j} (t_{j+1} - t_j) \\ \text{length}(\gamma) &= \int_a^b \,\left|\, \gamma'(t) \,\right|\, dt \qquad\\ \end{aligned}$$

!

Here is the definition of Integration with respect to Arc Length:

$$\begin{aligned} \int_\gamma f(z)\,\left|\, dz \,\right| = \int_a^b f(\gamma(t)) \,\left|\, \gamma'(t) \,\right|\,dt \end{aligned}$$

Note when $f(z) = 1$,

$$\begin{aligned} \int_\gamma f(z)\,\left|\, dz \,\right| = \int_\gamma \,\left|\, dz \,\right| = \int_a^b \,\left|\, \gamma'(t) \,\right|\,dt = \text{length}(\gamma) \end{aligned}$$

!

More at Analysis of a Complex Kind.

# posted by rot13(Unafba Pune) @ 8:28 AM