Residue | Complex Analysis | CSIR NET | SET | Gate Exams | Lec-18 | IFAS

Residue in Complex Analysis explained with clear concepts & exam-focused approach. Learn Residue definition, Laurent series, a₋₁ coefficient, pole residues & Cauchy residue theorem. For CSIR NET Mathematics, GATE Mathematics & SET Mathematics. 🔥Join India's Best Coaching for CSIR NET and GATE Exams: https://ifasonline.com/csir-net/mathe... 📞 Need Guidance? Call/WhatsApp: 9172266888 ━━━━━━━━━━━━━━━━━━━━━━━━━ ⏳ Lecture Timeline: [00:00:00] – Introduction To Residue [00:25:09] – Residue Definition (a₋₁ coefficient) [00:26:04] – Formula: Cauchy Integral for a₋₁ [00:27:32] – Question 1: Laurent series of 1/[z²(z-1)] [00:30:37] – Residue at 0 = -1 [00:30:43] – Question 2: Residue of e^z/z at 0 [00:32:29] – Residue = 1 [00:32:39] – Question 3: Residue of 1/[(z-1)(z-2)] [00:35:30] – Residue at 1 = -1, at 2 = 1 [00:36:24] – Residue at z = i (partial fractions) [00:37:52] – Integral = 6πi ⇒ Residue = 3 [00:41:36] – Question 4: Residue of sin z / z⁴ at 0 [00:42:47] – Residue = -1/6 [00:43:45] – Question 5: Functions with residue zero [00:45:45] – Trick: Even function → residue = 0 [00:47:34] – Residue of 1/(z²+1) at z = i [00:48:17] – Residue = 1/(2i) ⇒ Integral = π [00:49:49] – Residue of e^{iz}/(z²+a²) at ia [00:51:58] – Residue = e^{-a}/(2ia) [00:53:00] – Question 6: z²/sin z at z=0 [00:54:27] – Removable singularity → residue 0 [00:57:01] – Complete Session (Residue) ━━━━━━━━━━━━━━━━━━━━━━━━━ 📖 Topics Covered in This Lecture: Residue definition (a₋₁), Laurent series expansion, Residue at simple poles, Residue formula for higher poles, Integration via residue theorem, & Examples: 1/(z²(z-1)), e^z/z, sinz/z⁴ ━━━━━━━━━━━━━━━━━━━━━━━━━ 📚 Lecture Summary: In this session, IFAS Expert Faculty focuses on Residue in complex analysis. Residue = coefficient a₋₁ in Laurent series expansion of f(z) about isolated singularity. Key formula: ∫_C f(z) dz = 2πi × Res(f, z₀). For simple pole at z₀, Res = lim(z→z₀) (z−z₀)f(z). Worked examples: f(z)=1/(z²(z−1)) gives Res(0)=−1; f(z)=e^z/z gives Res(0)=1; sinz/z⁴ gives Res(0)=−1/6. Also covered: removable singularities give residue zero, even functions yield zero residue. Emphasizes exam problems for CSIR NET Mathematics, GATE Mathematics & SET Mathematics. ━━━━━━━━━━━━━━━━━━━━━━━━━ 🎓 This lecture is highly beneficial for students preparing for: ✅ CSIR NET Mathematical Science ✅ GATE Mathematics ✅ SET Mathematics ✅ TIFR Mathematics ✅ PhD Entrance Exams ━━━━━━━━━━━━━━━━━━━━━━━━━ 🏆 Why IFAS is India’s No. 1 Choice for CSIR NET & GATE Mathematics? (Our Results Speak): ✅ CSIR NET Dec 2025: 4000+ Selections (AIR 3, 8, 9.....) ✅ CSIR NET June 2025: 4200+ Selections (AIR 1, 2, 6, 10...) ✅ CSIR NET Dec 2024: 4000+ Selections (AIR 5, 6, 7, 8, 9...) ✅ CSIR NET June 2024: 2000+ Selections (AIR 3, 10, 13...) ✅ CSIR NET Dec 2023: 1200+ Selections (AIR 2, 4, 5...) 📄 View IFAS Result Sheet: 👉 https://drive.google.com/file/d/1THh6... ━━━━━━━━━━━━━━━━━━━━━━━━━ 🔗 Important Links: 🌍 Visit our Website: https://ifasonline.com 📞 Contact for Queries: 9172266888 📘 Free Counselling: https://ifasonline.com/csir-net/mathe... ━━━━━━━━━━━━━━━━━━━━━━━━━ 🔗 India's Best Online Classes and Books: 🔥CSIR NET Mathematical Science Live & Recorded Course: https://ifasonline.com/csir-net/mathe... 🎯 GATE Mathematics Recorded Course: https://ifasonline.com/gate/mathemati... 📘CSIR NET Mathematical Science Books: https://ifasonline.com/csir-net/mathe... ━━━━━━━━━━━━━━━━━━━━━━━━━ 📲 Download IFAS App: Android: https://play.google.com/store/apps/de... IOS: https://apps.apple.com/us/app/ifas/id... ━━━━━━━━━━━━━━━━━━━━━━━━━ Stay Updated Via Other Social Media: ✔Telegram: https://t.me/NET_GATE_SET_Maths_Discu... ✔Instagram:   / csir_net_mathematical_science   #residue #complexanalysis #csirnetmathematicalscience #gatemathematics #csirnetmaths #csirnetjune2026 #csirnetlecture #ifasmathematics