Types On Singularity | Complex Analysis | CSIR NET | SET | GATE Mathematics | Lec-6 | IFAS

Types On Singularity | Complex Analysis | CSIR NET | SET | GATE Mathematics | Lec-6 | IFAS explained with clear concepts and exam-focused approach. Master removable, pole, essential & isolated singularities with limit‑test tricks, pole‑order shortcuts, and PYQ‑based problem‑solving for CSIR NET, SET, and GATE 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 of Types On Singularity | Complex Analysis [00:02:30] – Recap: Isolated & Non-Isolated Singularity [00:05:40] – Isolated Singularity: 3 Types [00:07:09] – Concept: Removable Singularity [00:08:37] – Question 1: sin(z)/z at z=0 [00:12:16] – Trick: Finite Limit = Removable [00:14:26] – Question 2: (e^z -1)/z at z=0 [00:15:48] – Question 3: (z²-1)/(z-1) Redefined [00:19:04] – Important: Analytic = No Singularity [00:20:21] – Question 4: (1-cos z)/z² [00:22:40] – Question 5: Rational Function Simplification [00:25:42] – Question 6: log(1+z)/z at z=0 [00:29:21] – Concept: Pole Definition [00:31:00] – Example: sin(z)/z² Pole Order 1 [00:37:24] – Trick: Multiply by (z-a)^k [00:39:18] – Question 7: Pole of Order 3 [00:44:23] – Question 8: Poles of 1/(z²(z-2)) [00:53:02] – Question 9: Poles at ±i [00:54:43] – Question 10: (z²+1)/((z-1)(z+2)) [00:58:39] – Question 11: 1/(z²+1)² Poles [01:00:38] – Question 12: (e^z -1)/(z²(z-π)) [01:06:54] – Concept: Essential Singularity [01:07:22] – Trick: Limit Exists Nowhere [01:10:00] – Example: e^(1/z) Essential [01:10:45] – Example: sin(1/z) Essential [01:11:27] – Important: Non-Isolated Essential [01:19:55] – Complete Session (Types On Singularity | Complex Analysis ) ━━━━━━━━━━━━━━━━━━━━━━━━━ 📖 Topics Covered in This Lecture: • Isolated vs non‑isolated singularities • Removable singularities & finite limit test • Poles – order determination with (z‑a)^m f(z) • Essential singularities – limit exists neither • e^(1/z), sinz/z, 1/sin(1/z) examples • PYQ‑focused problem‑solving tricks ━━━━━━━━━━━━━━━━━━━━━━━━━ 📚 Lecture Summary: In this session, IFAS Expert Faculty focuses on the classification of singularities in complex analysis. The limit‑based approach distinguishes removable singularities (finite limit), poles (|f|→∞ with algebraic order), and essential singularities (limit neither finite nor infinite). Pole order is determined by finding the smallest m such that (z‑a)^m f(z) yields a non‑zero finite limit. Multiple examples and PYQ‑style questions reinforce the identification of singularity types, building a strong problem‑solving foundation for CSIR NET, SET, and GATE. ━━━━━━━━━━━━━━━━━━━━━━━━━ 🎓 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   #removablesingularity #polesingularity #essentialsingularity #isolatedsingularity #sinzoverz #1oversin1overz #typesonsingularity #complexanalysis #csirnetmathematicalscience #gatemathematics #csirnetmaths #csirnetjune2026 #csirnetlecture #ifasmathematics