GPT-4 Conversation #1101932128427581520

[tired heron]

Solve the contour integral (z+2)/(z^3-3z^2+4z-12)dz where C is a circle with r=2 and its center is 3+i and it travels counterclockwise.

[fierce oriole]

Let's find the singular points (poles) of the function, which are the solutions of the denominator to be 0.

z^3 - 3z^2 + 4z - 12 = 0

This factors to:

(z-1)(z-2)(z-6) = 0

So the singular points are z = 1, 2, and 6. Now we need to determine if any of these points lie inside the given contour.

The circle C has radius 2 and is centered at z₀ = 3 + i. In order for a pole to lie inside the contour, its distance to the center of the circle must be less than the radius. Let's check them:

Distance between z₀ and z = 1:

|1 - (3 + i)| = sqrt((-2)^2 + (-1)^2) = sqrt(5)

Distance between z₀ and z = 2:

|2 - (3 + i)| = sqrt((-1)^2 + (-1)^2) = sqrt(2)

Distance between z₀ and z = 6:

|6 - (3 + i)| = sqrt((3)^2 + (-1)^2) = sqrt(10)

The only pole within the contour of the circle C is z = 2.

Our function can be rewritten as:

f(z) = (z + 2) / [(z - 1)(z - 2)(z - 6)]

Since we're only concerned with the pole z = 2, let's perform partial fraction decomposition for this term only:

A / (z - 2)

To solve for A, multiply both sides by (z - 2), then set z = 2:

A = (2 + 2) / [(2 - 1)(2 - 6)] = 4 / (-4) = -1

So, our residue is -1. Now, we can find the value of the contour integral using the residue theorem:

Contour Integral = 2 * pi * i * (-1) = -2πi

The contour integral for the given function over the circle C is -2πi.

[tired heron]

Check if (z-1)(z-2)(z-6) are the correct factors.