Let \(f\) be a continuous function in the region \[ D = \{z \mid \abs{z} > R,\ 0 \le \arg z \le \theta\} \quad \text{where} \quad 1 \le \theta \le 2\pi. \] If there exists \(k\) such that \(\lim_{z\to\infty} zf(z) = k\) for \(z\) in the region \(D\). Show that \[ \lim_{R'\to\infty} \int_L f(z)\,dz = i\theta k, \] where \(L\) is the part of the circle \(\abs{z} = R'\) which lies in the region \(D\).
Limit of integrals over large circular arcs when $zf(z)\to k$
P-TIE-F09-02
- Area
- Complex Analysis
- Status
- draft