Complex Methods | Part IB, 2003


f(z)=zm1+zn,n>m+1,m,nN,f(z)=\frac{z^{m}}{1+z^{n}}, \quad n>m+1, \quad m, n \in \mathbb{N},

and let CRC_{R} be the boundary of the domain

DR={z=reiθ:0<r<R,0<θ<2πn},R>1.D_{R}=\left\{z=r e^{i \theta}: 0<r<R, \quad 0<\theta<\frac{2 \pi}{n}\right\}, \quad R>1 .

(a) Using the residue theorem, determine

CRf(z)dz\int_{C_{R}} f(z) d z

(b) Show that the integral of f(z)f(z) along the circular part γR\gamma_{R} of CRC_{R} tends to 0 as RR \rightarrow \infty.

(c) Deduce that

0xm1+xndx=πnsinπ(m+1)n\int_{0}^{\infty} \frac{x^{m}}{1+x^{n}} d x=\frac{\pi}{n \sin \frac{\pi(m+1)}{n}}

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