Let z be a complex variable. For a counter-clockwise integration around a unit circle C , centred at origin,  the value of A is ________________-

The correct option is 2/5.

Step-by-Step Explanation:

From the first provided image, we are given the contour integral:
$${\oint }_C\frac{1}{5z-4}dz=A\pi i$$
where $C$ is the unit circle, centered at the origin, oriented in the counter-clockwise direction (meaning $\left|z\right|=1$).

1. Identify the singularity (pole) of the integrand:
Let the integrand be:
$$f\left(z\right)=\frac{1}{5z-4}$$
To find the singularities, we set the denominator to zero:
$$5z-4=0\Rightarrow z=\frac{4}{5}$$
Thus, there is a simple pole at $z=\frac{4}{5}$.

2. Determine if the pole lies inside the contour C:
The contour $C$ is the unit circle $\left|z\right|=1$.
Since the absolute value of our pole is:
$$\left|z\right|=\left|\frac{4}{5}\right|=0.8<1$$
the pole $z=\frac{4}{5}$ lies strictly inside the unit circle $C$.

3. Evaluate the integral using Cauchy's Integral Formula:
As illustrated in the second image, we can rewrite the integrand by factoring out 5 from the denominator:
$${\oint }_C\frac{1}{5z-4}dz=\frac{1}{5}{\oint }_C\frac{1}{z-\frac{4}{5}}dz$$
Using Cauchy's Integral Formula, which states that for a function analytic inside and on $C$:
$${\oint }_C\frac{1}{z-z_0}dz=2\pi i$$
Substituting $z_0=\frac{4}{5}$ into the formula gives:
$${\oint }_C\frac{1}{z-\frac{4}{5}}dz=2\pi i$$
Therefore, the value of the complete integral is:
$${\oint }_C\frac{1}{5z-4}dz=\frac{1}{5}\left(2\pi i\right)=\frac{2}{5}\pi i$$

4. Determine the value of A:
Comparing the calculated result with the given equation from the second image:
$$A\pi i=\frac{2}{5}\pi i$$
Solving for $A$ yields:
$$A=\frac{2}{5}$$