Complex integral

Calculus Level 3

$\large \int \limits_{|z|=2} \frac {z-1}{(9+z^2)(x+i)} dz$

What is the value of the integral above, where $z$ is a complex number and $i=\sqrt{-1}$ denotes the imaginary unit ?

\frac{π}{4}(1-i)

\frac{π}{4}(1+i)

6πi

\frac{π}{4}

2πi

1 solution

Chew-Seong Cheong
Dec 11, 2018

Relevant wiki: Cauchy Integral Formula

Since $z=-i$ is the only singularity and $f(z) = \dfrac {z-1}{9+z^2}$ within the contour $|z| = 2$ . we can apply Cauchy integral formula as follows.

$\begin{aligned} I & = \int \limits_{|z|=2} \frac {z-1}{(9+z^2)(z+i)} dz = 2\pi 1 f(-i) = 2\pi i \left(\frac {-i-1}{9-1} \right) = \boxed{\dfrac \pi 4 (1-i)} \end{aligned}$

Complex integral

1 solution

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