Let C represent the unit circle centered at origin in the complex plane, and complex variable, z = x + iy. The value of the contour integral ${\oint }_C^{}\frac{\cosh 3z}{2z}dz$ (where integration is taken counter clockwise) is

The correct answer is πi.

To evaluate the contour integral:

$${\oint }_C\frac{\cosh (3z)}{2z}dz$$

we can use Cauchy's Integral Formula.

Cauchy's Integral Formula states that if a function f(z) is analytic within and on a simple closed contour C, and z₀ is any point inside C, then:

$${\oint }_C\frac{f(z)}{z-z_0}dz=2\pi if(z_0)$$

where the integration along C is taken in the counter-clockwise direction.

Here, the contour C is the unit circle centered at the origin, defined by |z| = 1.

Let us rewrite the given integral by separating the constant from the denominator:

$${\oint }_C\frac{\cosh (3z)}{2z}dz=\frac{1}{2}{\oint }_C\frac{\cosh (3z)}{z-0}dz$$

By comparing this to Cauchy's Integral Formula, we define the numerator as f(z) = cosh(3z) and the singularity point as z₀ = 0.

The point z₀ = 0 lies inside the unit circle contour C since its distance from the origin is 0, which is less than the radius of 1. Additionally, the function f(z) = cosh(3z) is analytic everywhere in the complex plane.

Evaluating f(z) at the singular point z₀ = 0:

$$f(0)=\cosh (3\cdot 0)=\cosh (0)=1$$

Substituting this result back into the integral calculation:

$$\frac{1}{2}{\oint }_C\frac{\cosh (3z)}{z-0}dz=\frac{1}{2}\cdot [2\pi i\cdot f(0)]=\frac{1}{2}\cdot 2\pi i\cdot 1=\pi i$$