Real Analysis & Complex Analysis | Target CSIR NET | MH SET | RJ SET | L-5 | IFAS

Real Analysis & Complex Analysis | Target CSIR NET | MH SET | RJ SET | L-5 | IFAS explained with clear concepts and exam-focused approach. Master Real Analysis and Complex Analysis PYQs for CSIR NET, MH SET, and RJ SET with expert problem‑solving. This lecture covers essential singularities, residue calculation tricks, harmonic conjugate problems, and entire function bounds – must‑know concepts for CSIR NET, GATE, and all SET exams. 🔥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 Real Analysis & Complex Analysis [00:05:43] – Question (z² cos(1/z) Singularity) [00:09:43] – Concept: Laurent Series & Essential Singularity [00:11:17] – Question (g(z)=(e^z‑1)/f(z) Poles) [00:13:51] – Concept: Zero Order & Pole Order [00:23:06] – Question (Entire Function Conditions) [00:27:36] – Question (Harmonic Conjugate) [00:30:52] – Trick: Write f(z) in x,y to find u,v [00:32:57] – Question (1/(z² sinh z) Singularities) [00:35:18] – Concept: Pole Order & Residue Calculation [00:45:33] – Question (Entire Function Growth Condition) [00:57:44] – Question (e^(1/z²) Properties) [01:03:17] – Question (Partial Derivatives at Origin) [01:08:25] – Question (Cardinality of Solution Set S) [01:11:02] – Trick: Define g(x)=e^(‑x)f(x) [01:14:43] – Question (Continuity with Parameter α) [01:17:24] – Trick: Polar Coordinates for Limit [01:20:00] – Question (Sum Σ 1/[n(n²‑1)]) [01:26:42] – Question (Limit of f(x) from f’) [01:41:12] – Concept: 1‑1 + Continuous → Strictly Monotone [01:46:42] – Question (Limit of Floor Function Sum) [01:49:16] – Trick: Bounds for Floor & Squeeze [01:54:57] – Question (Telescoping Series with Factorials) [02:06:56] – Complete Session (Real Analysis & Complex Analysis) ━━━━━━━━━━━━━━━━━━━━━━━━━ 📖 Topics Covered in This Lecture: • Singularity Classification: Essential Singularity, Poles, & Removable Singularity PYQs • Laurent Series Expansion & Residue Determination (1/z² sinhz, e^(1/z²)) • Harmonic Conjugates & Analytic Function Construction (Milne‑Thomson Method) • Entire Functions with Bounds & Polynomial Identification Tricks • Complex Integration & Residue Theorem Application Problems • Real Analysis: Limits, One‑One Functions, Greatest Integer Function • Infinite Series Summation (Telescoping & Factorial Series) • Fixed Points, Continuous & Strictly Monotonic Function Properties • Limit of Sequences using Squeeze Theorem & Inequalities ━━━━━━━━━━━━━━━━━━━━━━━━━ 📚 Lecture Summary: IFAS Expert Faculty covers Real Analysis & Complex Analysis PYQs for CSIR NET, MH SET, RJ SET. Master singularities (essential, poles), Laurent series, residue tricks, harmonic conjugates, entire functions. Real Analysis: limits, one‑one functions, greatest integer function, telescoping series. Time‑saving tricks with polar coordinates & Cauchy‑Riemann. Exam‑ready practice – Watch now! ━━━━━━━━━━━━━━━━━━━━━━━━━ 🎓 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   #realanalysis #complexanalysis #targetcsirnet #mhset #rjset #csirnetmathematicalscience #gatemathematics #csirnetmaths #csirnetjune2026 #csirnetlecture #ifasmathematics