compute Z= integral (z −a)^n dz where |z-a|=r is the countour where n is a given integer

To compute the integral Z = ∫(z - a)^n dz, where |z - a| = r is the contour and n is a given integer, we can follow these steps:

Step 1: Parameterize the contour |z - a| = r. Let z = a + re^(iθ), where θ is the parameter ranging from 0 to 2π.

Step 2: Substitute the parameterization into the integral. We have dz = ire^(iθ) dθ. The integral becomes Z = ∫(a + re^(iθ) - a)^n ire^(iθ) dθ.

Step 3: Simplify the expression inside the integral. (a + re^(iθ) - a)^n = (re^(iθ))^n = r^n e^(inθ).

Step 4: Substitute the simplified expression back into the integral. Z = ∫r^n e^(inθ) ire^(iθ) dθ.

Step 5: Simplify the expression further. Z = ir^(n+1) ∫e^(i(n+1)θ) dθ.

Step 6: Evaluate the integral. The integral of e^(i(n+1)θ) dθ is (1/(i(n+1))) e^(i(n+1)θ).

Step 7: Substitute the evaluated integral back into the expression. Z = ir^(n+1) (1/(i(n+1))) e^(i(n+1)θ) + C, where C is the constant of integration.

Step 8: Simplify the expression. Z = r^(n+1)/(n+1) e^(i(n+1)θ) + C.

Step 9: Convert back to the original variable. Since z = a + re^(iθ), we have e^(i(n+1)θ) = e^(i(n+1)(arg(z-a))).

Step 10: Finalize the expression. Z = r^(n+1)/(n+1) e^(i(n+1)(arg(z-a))) + C.

This is the step-by-step process to compute the integral Z = ∫(z - a)^n dz, where |z - a| = r is the contour and n is a given integer.