Evaluate the given integral, where C is the circle with positive orientation.

Cauchy integral theorem,

integral over C (2z-3)/(z^(2)-4)(z+2) dz, C:|z+3|=3

To evaluate the given integral using the Cauchy integral theorem, we need to proceed as follows:

1. Identify the pole(s) of the integrand within the contour C:|z+3|=3. A pole is a point where the denominator of the integrand becomes zero.

2. Find the residues at each pole. Residue is the coefficient of the term with the $(z−a)^{-1}$ in the Laurent series expansion of the integrand at each pole.

3. Apply the Cauchy residue theorem to calculate the integral.

Let's break it down step by step:

Step 1: Identify the poles
The integrand has potential poles at z=2 and z=-2, as they make the denominator equal to zero. However, since we are given that the contour C has a positive orientation, only the pole at z=-2 lies within C.

Step 2: Calculate the residue at z=-2
To calculate the residue at a simple pole, we can use the formula:
Res(f, -2) = lim(z→-2) ((z+2)(2z-3))/((z^2-4)(z+2))

Simplifying the expression inside the limit, we get:
Res(f, -2) = lim(z→-2) (2z-3)/(z^2-4)

Evaluating the limit gives:
Res(f, -2) = (2(-2)-3)/((-2)^2-4) = -7/2

Step 3: Apply the Cauchy residue theorem
According to the Cauchy residue theorem, the integral over C of the given function is equal to 2πi times the sum of the residues inside the contour C.

Therefore, the integral over C (2z-3)/(z^2-4)(z+2) dz = 2πi * (-7/2) = -7πi

So, the evaluated integral is -7πi.

To evaluate the given integral using the Cauchy integral theorem, we first need to check if the function (2z-3)/(z^2 - 4)(z+2) is analytic within the region enclosed by the circle C.

Analyticity is determined by the presence of singularities within the region of interest. Let's examine the function's denominator, (z^2 - 4)(z+2).

The singularities occur when the denominator is equal to zero. Therefore, we solve the equation z^2 - 4 = 0 and z + 2 = 0.

For z^2 - 4 = 0:
(z - 2)(z + 2) = 0,
z = 2, -2.

For z + 2 = 0:
z = -2.

These values indicate three singularities: z = 2, z = -2, and z = -2.

Based on the given condition |z+3| = 3, we can see that the circle C encloses the point z = -2. This means that the function has a singularity within the region enclosed by C.

Therefore, we cannot directly apply the Cauchy integral theorem to evaluate the given integral over C.

If you have any other questions, feel free to ask!