--- title: "Laurent series of 1/((1+z²)(z+2)), 1<|z|<2" subject: "Applied Mathematics" chapter: 1 key: "semester_4/applied_mathematics:laurent-1-over-1-z2-z-2" source: https://www.bctengineer.com/q/semester_4/applied_mathematics/laurent-1-over-1-z2-z-2 site: BCT Engineering --- # Laurent series of 1/((1+z²)(z+2)), 1<|z|<2 Applied Mathematics · Chapter 1: Complex Analysis - Asked in: 2069 Bhadra - Video: [complex analysis Taylor's and Laurent's series for function f(z) =1/(1+z^2)(z+2) || complex integral](https://www.youtube.com/watch?v=ngbhI20Hhwg) — Swati Theng Mathematics - 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-over-1-z2-z-2 - Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1 ## Question Obtain the Laurent series which represents the function f(z) = 1/((1 + z²)(z + 2)) when 1 < |z| < 2. ## Short answer In the annulus 1<|z|<2, f(z) = (1/5)[ Σ_{n=0}^∞ (−1)ⁿ zⁿ/2ⁿ⁺¹ + Σ_{n=0}^∞ (−1)ⁿ (2 z^{−2n−2} − z^{−2n−1}) ], i.e., (1/5)(… + 2/z² − 1/z + 1/2 − z/4 + …). ## Answer f(z) = 1/((1 + z²)(z + 2)), region 1 < |z| < 2. **Partial fractions:** 1/((z² + 1)(z + 2)) = A/(z + 2) + (Bz + C)/(z² + 1). 1 = A(z² + 1) + (Bz + C)(z + 2). Putting z = −2 gives A = 1/5. Comparing z²: A + B = 0, so B = −1/5. Comparing constants: A + 2C = 1, so C = 2/5. f(z) = (1/5) [ 1/(z + 2) + (2 − z)/(z² + 1) ] In the ring 1 < |z| < 2: |z| > 1 so expand in 1/z for the z² + 1 factor, and |z| < 2 so expand in z/2 for the z + 2 factor. **1/(z + 2)** (|z/2| < 1): 1/(z + 2) = (1/2) · 1/(1 + z/2) = (1/2) Σ (n = 0 to ∞) (−1)ⁿ zⁿ/2ⁿ = 1/2 − z/4 + z²/8 − … **(2 − z)/(z² + 1)** (|1/z²| < 1): 1/(z² + 1) = (1/z²) · 1/(1 + 1/z²) = Σ (n = 0 to ∞) (−1)ⁿ z^(−2n−2) = 1/z² − 1/z⁴ + 1/z⁶ − … so (2 − z)/(z² + 1) = −1/z + 2/z² + 1/z³ − 2/z⁴ − 1/z⁵ + 2/z⁶ + … = Σ (n = 0 to ∞) (−1)ⁿ [ 2 z^(−2n−2) − z^(−2n−1) ] **Answer:** f(z) = (1/5) { Σ (n = 0 to ∞) (−1)ⁿ zⁿ / 2ⁿ⁺¹ + Σ (n = 0 to ∞) (−1)ⁿ [ 2 z^(−2n−2) − z^(−2n−1) ] } = (1/5) [ … + 2/z² − 1/z + 1/2 − z/4 + z²/8 − … ] ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In 1 < |z| < 2, how is each factor expanded? A. 1/(z+2) in powers of 1/z; 1/(z²+1) in powers of z² B. 1/(z+2) in powers of z/2; 1/(z²+1) in powers of 1/z² C. Both in powers of z D. Both in powers of 1/z Answer: B. |z|<2 allows z/2; |z|>1 allows 1/z². (Key point: Ring conditions) 2. What is A in A/(z+2) + (Bz+C)/(z²+1)? A. A = -1/5 B. A = 1/3 C. A = 1/5 D. A = 2/5 Answer: C. Putting z=-2 gives 1 = A(5), so A = 1/5. (Key point: Partial fraction constant) 3. Which is the correct combined form of f(z)? A. (1/5)[1/(z+2) + (z-2)/(z²+1)] B. (1/5)[1/(z+2) - (2-z)/(z²+1)] C. (1/5)[1/(z+2) + (2+z)/(z²+1)] D. (1/5)[1/(z+2) + (2-z)/(z²+1)] Answer: D. B = -1/5 and C = 2/5 give (2 - z)/5 over z²+1. (Key point: Decomposed form) ## Close variants set in other exams - [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× ## 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 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 - [Laurent series of (1-cos z)/z³](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Alaurent-1-cosz-over-z3) — Applied Mathematics, chapter 1 ## 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 - [Fourier series of a function](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afourier-series-of-a-function) — Mathematics III, chapter 4, asked 6× - [Valid lossless function; Foster series and Cauer I](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Alossless-four-functions-foster-series) — Filter Design, chapter 4, asked 2× - [Series programming of a third-order transfer function](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aseries-programming-z-plus-1) — Digital Control System, chapter 4, asked 2×