The closed loop line integral _≤ft| z | = 5^ {z^3 + z^2 + 8}z + 2dz evaluated counter-clockwise, is

Complex Contour Integral Evaluation

This solution demonstrates the step-by-step evaluation of the specified closed loop line integral using fundamental concepts from complex analysis.

Integral and Contour Details

The problem requires evaluating the following integral:

_≤ft| z | = 5^ {z^3 + z^2 + 8}z + 2dz

The integration path is the circle defined by $\left|z\right|=5$. This represents a circle centered at the origin (0,0) with a radius of 5 units in the complex plane. The integral traverses this contour in the counter-clockwise direction.

Singularity Identification

The integrand function is $f(z)=\frac{z^3+z^2+8}{z+2}$. Singularities occur where the function is undefined, typically when the denominator is zero.

By setting the denominator to zero:

$$z+2=0$$

We find a single singularity at $z=-2$.

Singularity Location Check

To apply Cauchy's Integral Formula, it's crucial to determine if the singularity lies within the specified contour.

The contour is the circle $\left|z\right|=5$.

The singularity is at $z=-2$.

We check the magnitude of the singularity: $\left|-2\right|=2$.

Since $2<5$, the condition $\left|-2\right|<5$ is met. Therefore, the singularity $z=-2$ is inside the contour.

Cauchy's Integral Formula Application

Cauchy's Integral Formula provides a direct method for calculating integrals of analytic functions around closed paths. The formula is stated as follows: If $g(z)$ is analytic (holomorphic) inside and on a simple closed contour $C$, and $a$ is any point strictly inside $C$, then:

$$\underset{C}{\oint }\frac{g(z)}{z-a}dz=2\pi ig(a)$$

For the given integral, we can match it to the formula by identifying:

- The function $g(z)$ as the part of the integrand that is analytic inside the contour: $g(z)=z^3+z^2+8$. This is a polynomial and thus analytic everywhere.
- The point $a$ corresponding to the singularity in the denominator: $z+2=z-(-2)$, so $a=-2$.

The conditions for Cauchy's Integral Formula are satisfied because $g(z)$ is analytic, and $a=-2$ lies inside the contour $\left|z\right|=5$.

Integral Value Calculation

First, calculate the value of $g(z)$ at the point $a=-2$:

$$g(-2)=(-2{)}^3+(-2{)}^2+8$$

Calculate the powers:

$$g(-2)=-8+4+8$$

Perform the addition:

$$g(-2)=4$$

Now, substitute $g(-2)$ into Cauchy's Integral Formula:

_≤ft| z | = 5^ {z^3 + z^2 + 8}z + 2dz = 2π i g(-2)

$$=2\pi i\times 4$$

$$=8\pi i$$

In engineering and some mathematical contexts, $j$ is used instead of $i$ to represent the imaginary unit. Therefore, the result can also be expressed as $8j\pi$.