--- title: "Evaluate ∮(4-3z)/(z(z-1)(z-2)) on |z|=3/2" subject: "Applied Mathematics" chapter: 1 key: "semester_4/applied_mathematics:cif-4-3z-over-z-z-1-z-2" source: https://www.bctengineer.com/q/semester_4/applied_mathematics/cif-4-3z-over-z-z-1-z-2 site: BCT Engineering --- # Evaluate ∮(4-3z)/(z(z-1)(z-2)) on |z|=3/2 Applied Mathematics · Chapter 1: Complex Analysis - Asked in: 2071 Magh, 5 marks; 2069 Bhadra; 2068 Magh, 5 marks (asked 3 times) - Video: [if C is a Circle of |z|=3/2 Evaluate integral 4-3z/z(z-1)(z-2) dz #complex analysis](https://www.youtube.com/watch?v=Q9eoGoxZw6E) — Maan Shree - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/cif-4-3z-over-z-z-1-z-2 - Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1 ## Question State Cauchy's integral formula and evaluate ∮_C (4 - 3z)/(z(z - 1)(z - 2)) dz, where C is the circle |z| = 3/2. ## Short answer For |z|=3/2 only z=0 and z=1 are inside. Partial fractions give 2/z − 1/(z−1) − 1/(z−2); the last term is analytic inside, so by CIF I=4πi−2πi+0=2πi. ## Answer **Cauchy's integral formula:** if f(z) is analytic inside and on a simple closed contour C, and *a* is a point inside C, then f(a) = (1/2πi) ∮_C f(z)/(z − a) dz, that is, ∮_C f(z)/(z − a) dz = 2πi f(a). #### Solution C: |z| = 3/2. The poles are z = 0, 1, 2. Since |0| < 3/2 and |1| < 3/2, but |2| > 3/2, the poles z = 0 and z = 1 are inside C, and z = 2 is outside. Partial fractions: (4 − 3z)/(z(z − 1)(z − 2)) = A/z + B/(z − 1) + C/(z − 2) ```text A = (4 − 0)/((0 − 1)(0 − 2)) = 4/2 = 2 B = (4 − 3)/(1·(1 − 2)) = 1/(−1) = −1 C = (4 − 6)/(2·(2 − 1)) = −2/2 = −1 ``` So the integrand is 2/z − 1/(z − 1) − 1/(z − 2). Apply CIF with f(z) = 1 for the first two terms. The last term, −1/(z − 2), is analytic inside C, so its integral is 0 by Cauchy's theorem. ```text ∮ 2/z dz = 2 · 2πi · 1 = 4πi ∮ −1/(z − 1) dz = −2πi ∮ −1/(z − 2) dz = 0 ``` I = 4πi − 2πi + 0 = 2πi **Answer: 2πi** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For |z| = 3/2, which poles of (4−3z)/(z(z−1)(z−2)) are inside? A. Only z = 0 B. z = 0, 1 and 2 C. z = 0 and z = 1 D. z = 1 and 2 Answer: C. |2| > 3/2 so z = 2 is outside. (Key point: Poles) 2. The partial fractions are: A. 1/z + 1/(z−1) + 1/(z−2) B. 2/z − 1/(z−1) − 1/(z−2) C. 2/z + 1/(z−1) − 1/(z−2) D. −2/z + 1/(z−1) + 1/(z−2) Answer: B. A=2, B=−1, C=−1. (Key point: Coefficients) 3. Why does −1/(z−2) contribute 0? A. CIF gives 2πi f(2) = 0 B. The residue is infinite C. Its coefficient is zero D. It is analytic inside C, so Cauchy's theorem gives 0 Answer: D. Pole 2 lies outside C. (Key point: Outside pole) 4. The value of the integral is: A. −2πi B. 2πi C. 4πi D. 0 Answer: B. 4πi − 2πi + 0 = 2πi. 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