\(f: D\rightarrow {\mathbb C}\) be a continuous function, where \(D\subset {\mathbb C}\) is a domain.Let \(\alpha:[a,b]\rightarrow D\) be a smooth curve. Give a precise definition of the complex line integral \[\int_{\alpha} f.\]
Assume that there exists a constant \(M\) such that \(|f(\tau)|\leq M\) for all \(\tau\in \mbox{\textrm Image}(\alpha\)). Prove that \[\big | \int_{\alpha} f \big |\leq M \times \mbox{\textrm length}(\alpha).\]
Let \(C_R\) be the circle \(|z|=R\), described in the counterclockwise direction, where \(R>1\). Using the parametrization \(z=Re^{it}\) for \(0\le t\le2\pi\) and the pathwise determination \(\log(Re^{it})=\log R+it\), provide an upper bound for \(\big | \int_{C_R} \dfrac{\log{(z)} }{z^2} \big |\) depending only on \(R\) and universal constants.
Solution
1. If \(\alpha:[a,b]\to D\) is piecewise \(C^1\), define\[ \int_\alpha f(z)\,dz :=\int_a^b f(\alpha(t))\alpha'(t)\,dt, \]where the right-hand side is the usual integral of a complex-valued function of a real variable.
2. If \(|f(\alpha(t))|\le M\) for all \(t\), then\[ \left|\int_\alpha f(z)\,dz\right| \le\int_a^b|f(\alpha(t))|\,|\alpha'(t)|\,dt \le M\int_a^b|\alpha'(t)|\,dt. \]Thus\[ \left|\int_\alpha f(z)\,dz\right| \le M\,\operatorname{length}(\alpha). \]
3. For part (c), a global continuous branch of \(\log z\) does not exist on the circle \(C_R\) because it winds once around \(0\). Interpret the integral along the standard parametrization\[ \alpha(t)=Re^{it},\qquad0\le t\le2\pi, \]using the pathwise determination\[ \log(\alpha(t))=\log R+it. \]Then\[ |\log(\alpha(t))| \le\log R+2\pi \]and\[ \left|\frac{\log(\alpha(t))}{\alpha(t)^2}\right| \le\frac{\log R+2\pi}{R^2}. \]Since \(\operatorname{length}(C_R)=2\pi R\), the estimate from 2 gives\[ \left|\int_{C_R}\frac{\log z}{z^2}\,dz\right| \le\frac{2\pi(\log R+2\pi)}{R} \]for this specified determination of the logarithm along the path.