--- title: "Laurent series of (1-cos z)/z³" subject: "Applied Mathematics" chapter: 1 key: "semester_4/applied_mathematics:laurent-1-cosz-over-z3" source: https://www.bctengineer.com/q/semester_4/applied_mathematics/laurent-1-cosz-over-z3 site: BCT Engineering --- # Laurent series of (1-cos z)/z³ Applied Mathematics · Chapter 1: Complex Analysis - Asked in: 2069 Poush - Video: [Find the Residue of a Function using Laurent Series and known Taylor Series](https://www.youtube.com/watch?v=FoTJ6SW2bRE) — sumchief - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/laurent-1-cosz-over-z3 - Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1 ## Question Find the Laurent's series expansion of f(z) = (1 - cos z)/z³, 0 < |z| < R. ## Short answer Using the Maclaurin series of cos z, f(z) = 1/(2z) − z/24 + z³/720 − …, valid for 0<|z|<∞. The principal part is 1/(2z), so z=0 is a simple pole and the residue is 1/2. ## Answer Use the Maclaurin series of cos z, which converges for all z: cos z = 1 − z²/2! + z⁴/4! − z⁶/6! + … so 1 − cos z = z²/2! − z⁴/4! + z⁶/6! − z⁸/8! + … Divide by z³: f(z) = (1 − cos z)/z³ = 1/(2z) − z/24 + z³/720 − z⁵/40320 + … **Answer:** f(z) = Σ (n = 1 to ∞) (−1)ⁿ⁺¹ z^(2n − 3) / (2n)! valid for 0 < |z| < ∞. - Principal part: 1/(2z) only, so z = 0 is a simple pole. - Residue at z = 0 is the coefficient of 1/z, which is **1/2**. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which series is used to expand (1-cos z)/z³? A. Maclaurin series of sin z B. Partial fractions of 1/z³ C. Maclaurin series of cos z, then subtract from 1 D. Binomial series of 1/(1-z) Answer: C. 1 - cos z = z²/2! - z⁴/4! + ..., then divide by z³. (Key point: Starting series) 2. What are the first terms of the Laurent series? A. 1/(2z) + z/24 + z³/720 + ... B. 1/z³ - 1/(2z) + ... C. 1/(2z²) - 1/24 + ... D. 1/(2z) - z/24 + z³/720 - ... Answer: D. z²/2 divided by z³ gives 1/(2z); -z⁴/24 gives -z/24. (Key point: Leading terms) 3. What is the nature of z = 0 and the residue? A. Simple pole; residue 1/2 B. Pole of order 3; residue 1/2 C. Simple pole; residue 1 D. Removable singularity; residue 0 Answer: A. Only 1/z appears in the principal part, with coefficient 1/2. (Key point: Singularity and residue) ## Related in Applied Mathematics - [Laurent series of 1/(z²-3z+2), 1<|z|<2](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alaurent-1-over-z2-3z-2) — Applied Mathematics, chapter 1, asked 2× - [Laurent series of (z²-1)/((z+2)(z+3)), 2<|z|<3](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alaurent-z2-1-over-z2-z3) — Applied Mathematics, chapter 1, asked 2× - [Laurent series of 1/((1+z²)(z+2)), 1<|z|<2](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alaurent-1-over-1-z2-z-2) — Applied Mathematics, chapter 1, asked 2× - [Laurent series of z/((z+1)(z+2)), 1<|z|<2](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alaurent-z-over-z1-z2) — Applied Mathematics, chapter 1 - [Laurent series of sin z / z⁶](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alaurent-sinz-over-z6) — Applied Mathematics, chapter 1 - [Taylor series of 1/(z²+4) about z = i](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Ataylor-series-1-over-z2-plus-4-about-i) — Applied Mathematics, chapter 1, asked 5× ## Related in other subjects - [Laurent series expansion](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaurent-series-expansion) — Mathematics III, chapter 2 - [Taylor series expansion](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ataylor-series-expansion) — Mathematics III, chapter 2 - [Taylor series expansion](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ataylor-series-expansion-two) — Mathematics III, chapter 2 - [Taylor series expansion](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ataylor-series-expansion-three) — Mathematics III, chapter 2