I'm doing a Complex Analysis course and I'm struggling on understanding which is the difference between $long(\gamma)$ and $long(f(\gamma))$, As far as I know, my guess is that $$long(\gamma)=\int_{\gamma}|dz|$$ and $$long(f(\gamma))=\int_{\gamma}f(z)|dz|$$
Is my assumption correct?
I would be really gratefull if someone could tell me if that is correct and, if not, explain me the correct way of computing this values. Thanks in advance!
When $$\gamma:\quad t\mapsto z(t)\qquad(a\leq t\leq b)\tag{1}$$ is a curve in the $z$-plane then its length, called by you ${\rm long}(\gamma)$, is given by $${\rm long}(\gamma)=\int_\gamma |dz|=\int_a^b |\dot z(t)|\>dt\ .$$ It seems that we are given in addition a holomorphic function $$f:\quad z\mapsto w=f(z)$$ whose domain includes $\gamma$. Then we can talk about the image curve $f(\gamma)$ in the $w$-plane and its length. From $(1)$ we obtain the following parametrization of $f(\gamma)$: $$f(\gamma):\quad t\mapsto w(t)=f\bigl(z(t)\bigr)\qquad(a\leq t\leq b)\ .$$ By definition $${\rm long}\bigl(f(\gamma)\bigr)=\int_{f(\gamma)}|dw|=\int_a^b|\dot w(t)|\>dt=\int_a^b\bigl|f'\bigl(z(t)\bigr)\bigr|\>|\dot z(t)|\>dt\ .$$ Here the factor $\bigl|f'\bigl(z(t)\bigr)\bigr|$ in the last integral encompasses the linear stretching of the line elements at $z(t)$ when $f$ is applied.