--- title: "Evaluate ∮(z+1)dz over a square" subject: "Applied Mathematics" chapter: 1 key: "semester_4/applied_mathematics:complex-integral-square-contour" source: https://www.bctengineer.com/q/semester_4/applied_mathematics/complex-integral-square-contour site: BCT Engineering --- # Evaluate ∮(z+1)dz over a square Applied Mathematics · Chapter 1: Complex Analysis - Asked in: 2069 Bhadra - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/complex-integral-square-contour - Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-1 ## Question Define complex integral. Evaluate: ∮_C (z + 1) dz where C is the square with vertices at z = 0, z = 1, z = 1 + i and z = i. ## Short answer Define ∮_ C f(z)dz = ∫(u dx−v dy)+i∫(v dx+u dy). For f(z)=z+1, entire and analytic on and inside the closed square with vertices 0,1,1+i,i. By Cauchy integral theorem, the integral is zero. Direct antiderivative F(z)=z^2/2+z gives same: F(end)−F(start)=0. ## Answer #### Complex integral For continuous f(z) on a curve C: ∫_C f(z) dz = ∫_C (u dx − v dy) + i ∫_C (v dx + u dy). #### Solution f(z) = z + 1 is a polynomial, so it is analytic everywhere, in particular inside and on the square C with vertices 0, 1, 1 + i, i. By **Cauchy's integral theorem**, ∮_C f(z) dz = 0. #### Verification by direct integration An antiderivative is F(z) = z²/2 + z. Along a closed path the start and end points coincide, so ∮_C (z + 1) dz = F(end) − F(start) = 0. Direct check, one side at a time (counter-clockwise): ```text 0 → 1 : ∫₀¹ (x + 1) dx = 3/2 1 → 1+i : z = 1 + iy, dz = i dy ∫₀¹ (2 + iy) i dy = 2i − 1/2 1+i → i : z = x + i, x: 1→0 ∫₁⁰ (x + 1 + i) dx = −(1/2 + 1 + i) = −3/2 − i i → 0 : z = iy, y: 1→0 ∫₁⁰ (iy + 1) i dy = −(−1/2 + i) = 1/2 − i ``` Real parts: 3/2 − 1/2 − 3/2 + 1/2 = 0. Imaginary parts: 2 − 1 − 1 = 0. **Answer: ∮_C (z + 1) dz = 0** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is the integral of (z+1) around the square equal to 0? A. z+1 is analytic inside and on C, so Cauchy's integral theorem applies B. z+1 has a pole inside the square with zero residue C. The square has equal sides so contributions cancel D. z+1 is real on the contour Answer: A. A polynomial is analytic everywhere, so Cauchy's theorem gives zero. (Key point: Why zero) 2. Which antiderivative shows the closed-path integral is F(end) - F(start) = 0? A. F(z) = z² + z B. F(z) = z²/2 + z C. F(z) = z²/2 + 1 D. F(z) = 2z + 1 Answer: B. d/dz(z²/2 + z) = z + 1; on a closed path start = end so the difference is 0. (Key point: Antiderivative check) 3. On the side from 1 to 1+i, how is the integral set up? A. z = x + i, dz = dx, x from 1 to 0 B. z = 1 + iy, dz = dy, y from 0 to 1 C. z = 1 + iy, dz = i dy, y from 0 to 1 D. z = iy, dz = i dy, y from 1 to 0 Answer: C. On that vertical side x = 1 so z = 1 + iy and dz = i dy. (Key point: Side parametrisation) ## Related in Applied Mathematics - [Define complex integral; evaluate ∮log z dz](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acomplex-integral-log-z) — Applied Mathematics, chapter 1 - [Evaluate ∫(x²+iy)dz along y = x²](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acomplex-integral-path-y-x2) — Applied Mathematics, chapter 1 - [Evaluate ∫(x²+iy)dz along y = x](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acomplex-integral-path-y-x) — Applied Mathematics, chapter 1 - [Cauchy theorem vs formula; evaluate integral](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Acif-distinguish-e-z-z-1-z-2) — Applied Mathematics, chapter 1, asked 4× - [Residue theorem: ∮(1-cos z)/z³ dz](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aresidue-1-cosz-over-z3) — Applied Mathematics, chapter 1 ## Related in other subjects - [Evaluate a line integral](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-a-line-integral-three) — Mathematics III, chapter 2 - [Evaluate a line integral F=sinyi+x(1+cosy)j](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-a-line-integral-f-sinyi) — Mathematics III, chapter 2, asked 6× - [Triple integral x² over tetrahedron](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Atriple-integral-x2-over-tetrahedron) — Mathematics, chapter 2, asked 2× - [Evaluate a line integral](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-a-line-integral-two) — Mathematics III, chapter 2