Evaluate HC2z+3z dz where C is upper half of the circle |z| = 2 in clockwise direction

To solve this integral, we will use the Cauchy Integral Formula. The Cauchy Integral Formula states that for a function f(z) that is analytic inside and on a simple closed contour C and for any point z0 inside C, the integral over C of f(z)/(z-z0) dz is 2πif(z0).

First, we rewrite the integral in terms of the Cauchy Integral Formula. We have:

∫C (2z + 3z) dz = ∫C 5z dz

We can see that this is in the form of the Cauchy Integral Formula with f(z) = 5z and z0 = 0. Therefore, by the Cauchy Integral Formula, the integral is 2πi * f(0) = 2πi * 5 * 0 = 0.

However, we are given that the contour C is traversed in the clockwise direction. The Cauchy Integral Formula assumes a counter-clockwise orientation. Therefore, we must multiply our result by -1 to account for the opposite orientation.

So, the value of the integral ∫C (2z + 3z) dz over the given contour C is -1 * 0 = 0.