The value of the contour integral in the complex plane

$\oint \frac{z^3-2z+3}{z-2}dz$

along the contour |z| = 3, taken counterclockwise is

Evaluating Contour Integral Value Using Cauchy's Formula

We are asked to evaluate the contour integral:

$${\oint }_C\frac{z^3-2z+3}{z-2}dz$$

Where the contour $C$ is defined by $|z|=3$, taken in the counterclockwise direction.

Identifying the Function and Contour

The integral is of the form ${\oint }_C\frac{g(z)}{h(z)}dz$. Here, the integrand is $g(z)=\frac{z^3-2z+3}{z-2}$. The contour $C$ is the circle centered at the origin with radius 3.

The denominator $z-2$ indicates a potential singularity (a pole) at $z=2$. We need to check if this singularity lies inside the contour $C$. The condition $|z|=3$ describes a circle of radius 3 centered at the origin. Since $|2|=2$, which is less than 3, the point $z=2$ lies inside the contour $C$.

Applying Cauchy's Integral Formula

Cauchy's Integral Formula is a powerful tool for evaluating contour integrals. It states that if a function $f(z)$ is analytic inside and on a simple closed contour $C$, and $a$ is any point inside $C$, then:

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

Applying the Formula to the Problem

In our specific integral, ${\oint }_C\frac{z^3-2z+3}{z-2}dz$, we can identify:

- $f(z)=z^3-2z+3$
- $a=2$
- $C$ is the circle $|z|=3$

First, we must verify that $f(z)=z^3-2z+3$ is analytic inside and on the contour $C$. Since $f(z)$ is a polynomial, it is analytic everywhere in the complex plane. Therefore, it is analytic inside and on the circle $|z|=3$.

Calculation

Now, we apply Cauchy's Integral Formula:

1. Evaluate $f(a)$: Substitute $z=a=2$ into $f(z)$. $f(2)=(2{)}^3-2(2)+3$ $f(2)=8-4+3$ $f(2)=7$
2. Calculate the integral value: Multiply $f(2)$ by $2\pi i$. Integral Value = $2\pi i\times f(2)$ Integral Value = $2\pi i\times 7$ Integral Value = $14\pi i$

Conclusion

The value of the contour integral $\oint \frac{z^3-2z+3}{z-2}dz$ along the contour $|z|=3$, taken counterclockwise, is $14\pi i$.