Change of variables
by Hidenori
Proposition
Prove Proposition 4.2 (A first course in complex analysis) and the fact that the length of $\gamma$ does not change under reparametrization.
Solution
Let $\tau:[c, d] \rightarrow [a, b]$ be given such that $\sigma = \gamma \circ \tau$.
\[\begin{align*} \int_{c}^{d} f(\sigma(t))\sigma'(t) dt &= \int_{c}^{d} f((\gamma \circ \tau)(t))(\gamma \circ \tau)'(t) dt \\ &= \int_{c}^{d} ((f \circ \gamma) \cdot \gamma')(\tau(t))\tau'(t) dt \\ &= \int_{a}^{b} ((f \circ \gamma) \cdot \gamma')(t) dt & \text{(Theorem A.6)} \\ &= \int_{a}^{b} f(\gamma(t))\gamma'(t) dt. \end{align*}\] \[\begin{align*} \int_c^d \abs{\sigma'(t)} dt &= \int_c^d \abs{\gamma'(\tau(t))}\abs{\tau'(t)} dt \\ &= \int_c^d \abs{\gamma'(\tau(t))}\tau'(t) dt & \text{($\tau$ is increasing.)} \\ &= \int_a^b \abs{\gamma'(t)} dt & \text{(Theorem A.6)}. \end{align*}\]Subscribe via RSS