Question

Use the theorem in Sec. 77, involving a single residue, to evaluate the integral of f (z) around the positively oriented circle |z| 3 when intergral c f(z) dz, f(z) = (z^3(1-3z))/(1+z) (1+2z^4)

Answer 1

The integral of f (z) around the positively oriented circle |z| = 3 is equal to 2πi times the sum of the residues of the poles of f (z) inside the circle.

The poles of f (z) inside the circle are z = 0 and z = −1/2.

The residue at z = 0 is f (0) = 0.

The residue at z = −1/2 is f (−1/2) = −7/16.

Therefore, the integral of f (z) around the positively oriented circle |z| = 3 is equal to 2πi (−7/16) = −7πi/8.

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