The magnitude of the contour integral $\oint C_{}\frac{{(z+1)}^2}{(z-i)(z-2)}dz$  over the contour $C$ : $|z-2-i|=\frac{3}{2}$  is $\_\_\_\_\_\_$  (Round off to two decimal places)

The correct option is 25.29.

To find the magnitude of the contour integral, we can apply Cauchy's Residue Theorem.

Let the integrand be defined as:

$$f\left(z\right)=\frac{{(z+1)}^2}{(z-i)(z-2)}$$

The singularities (poles) of the integrand $f\left(z\right)$ are the values of $z$ where the denominator is zero:
1. A simple pole at $z=i$
2. A simple pole at $z=2$

The given contour $C$ is a circle described by the equation:

$$|z-(2+i\left)\right|=\frac{3}{2}=1.5$$

This is a circle centered at $z_0=2+i$ with a radius of $R=1.5$.

We check which of the poles lie inside this contour $C$:
- For the pole at $z=i$:
Distance to the center is $|i-(2+i\left)\right|=|-2|=2$.
Since $2>1.5$, the pole $z=i$ lies outside the contour $C$.
- For the pole at $z=2$:
Distance to the center is $|2-(2+i\left)\right|=|-i|=1$.
Since $1<1.5$, the pole $z=2$ lies inside the contour $C$.

By Cauchy's Residue Theorem, the value of the contour integral is given by:

$$\oint C_{}f\left(z\right)dz=2\pi i·{\text{Res}}_{z=2}f\left(z\right)$$

Next, we calculate the residue of $f\left(z\right)$ at the simple pole $z=2$:

$${\text{Res}}_{z=2}f\left(z\right)=\underset{z\to 2}{lim}(z-2)\frac{{(z+1)}^2}{(z-i)(z-2)}=\frac{{(2+1)}^2}{2-i}=\frac{9}{2-i}$$

Now, we substitute this residue back into the integral equation:

$$I=2\pi i·\frac{9}{2-i}=\frac{18\pi i}{2-i}$$

To find the magnitude of this complex number, we compute the absolute value:

$$\left|I\right|=\left|\frac{18\pi i}{2-i}\right|=\frac{18\pi ·\left|i\right|}{|2-i|}$$

Evaluating the magnitudes of the components:
- $\left|i\right|=1$
- $|2-i|=\sqrt{2^2+{(-1)}^2}=\sqrt{5}$

Substituting these values back into the equation:

$$\left|I\right|=\frac{18\pi }{\sqrt{5}}$$

Using the numerical values $\pi \approx 3.14159$ and $\sqrt{5}\approx 2.23607$:

$$\left|I\right|\approx \frac{18·3.14159}{2.23607}\approx \frac{56.54867}{2.23607}\approx 25.289$$

Rounding to two decimal places gives 25.29.