Question

The integral \(\oint f\left( z \right)dz\) evaluated around the unit circle on the complex plane for \(f\left( z \right) = \frac{{\cos z}}{z}\) is
1. 2 π i
2. 4 π i
3. -2 π i
4. 0

Answer 1

Correct Answer - Option 1 : 2 π i

f(z) \(= \frac{{cosZ}}{Z}\) has simple pole at Z = 0

∴ Residue of f(z) at Z = 0

\(\begin{array}{*{20}{c}} {Lt\;}\\ {z \to 0} \end{array}\;f\left( z \right).z = \begin{array}{*{20}{c}} {Lt\;cosz = 1\;}\\ {z \to 0} \end{array}\)

\(\mathop \smallint \limits_e^\; f\left( z \right)dz = 2\pi i\) (Residue at Z = 0)

= 2π i × 1 = 2π i