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Complex Analysis and its extension | Deeper Mathematical Insights

🎙 Talk of the Week 🌟 Complex Analysis and it's extensions : A deeper Mathematical Insight ✨ “Build your dreams with purpose, learn with passion, and grow with direction.” 📅 Date: 18th October 2025 🕒 Time: 06:00 PM 📺 Live on YouTube: [@prajnafoundation2025] 💬 “Building a strong future begins with clear goals, lifelong learning, and purposeful career choices — the true foundation for success.” Got it 👍 — you want to *create a clean YouTube video description with timestamps* using the transcript you provided. Below is a well-structured **YouTube summary with time stamps**, formatted in a professional way (perfect for the “Description” section of your video). Talk of the Week – “Complex Analysis and its Extensions: A Deeper Mathematical Insight” 🎙️ *Speaker:* Dr. Nagarjun V. 🏫 *Head, Department of Mathematics, The Oxford College of Science, Bengaluru* 🎓 *Organized by:* Prajna Foundation – Pragna Prerana: Talk of the Week 📅 *Session Chair:* Dr. Mahesh Arvin, Chairman, Prajna Foundation 00:00:08 – 00:02:27 🎵 Opening music and welcome address by **Dr. Mahesh Arvin**, Chairman of Prajna Foundation. He introduces the session topic — “Complex Analysis and its Extensions: A Deeper Mathematical Insight.” He also shares that the speaker, **Dr. Nagarjun V**, was once his student during undergraduate days at Vijaya College. 00:01:25 – 00:03:21 📘 Introduction to *Dr. Nagarjun V* — his academic background, Ph.D. in Nevanlinna Theory, 8+ years of teaching experience, and research contributions with over 15 papers and conference presentations. 00:02:27 – 00:05:50 🎤 *Dr. Nagarjun’s Talk Begins.* Brief introduction to *Complex Numbers* — origin, definition of algebraic and transcendental equations, and how imaginary numbers emerged. 00:05:37 – 00:09:22 📊 Evolution of *Complex Analysis* — contributions from Euler, Gauss, and Riemann. Applications in analytical number theory, conformal mapping, engineering, physical sciences, and fluid mechanics. 00:08:40 – 00:11:27 🧩 Explanation of the *complex number form* Z = X + iY, real and imaginary parts, and representation on the complex plane. Definition of *modulus* and basic algebra of complex numbers (addition, subtraction, multiplication, division). 00:11:33 – 00:15:59 📐 Concepts of *conjugate* and *polar form* of a complex number. Conversion between Cartesian and polar forms using r cos θ and r sin θ. Introduction to *modulus* as the distance from the origin in the complex plane. 00:16:00 – 00:20:54 📈 *Limit, continuity, and differentiability* in complex functions. How calculus of real numbers extends to complex numbers. Definition of *analytic (holomorphic) functions* and *harmonic functions* (Laplace equation). 00:20:59 – 00:24:00 📚 *Cauchy–Riemann equations* – conditions for differentiability and analyticity. Introduction to *Cauchy’s Theorem* for closed curves in simply connected regions. 00:23:18 – 00:25:26 🧮 *Cauchy’s Integral Formula* and *Morera’s Theorem* — relation between analytic functions and contour integrals. 00:24:47 – 00:29:41 ⚛️ *Singularities in Complex Functions* — definition and types: Isolated singularity Removable singularity (e.g., sin z / z at z = 0) Essential singularity Poles (simple, double, multiple) 00:29:41 – 00:31:38 🌐 *Entire and Meromorphic Functions* — differences, examples (sin z, cos z), and pole-based definitions. 00:30:59 – 00:33:14 📖 Introduction to *Nevanlinna (Value Distribution) Theory* — an extension of complex analysis developed by Finnish mathematician Rolf Nevanlinna in the 1920s. Explanation of *Poisson–Jensen Formula* and counting of zeros with multiplicity. 00:33:40 – 00:37:54 📊 Core definitions in Nevanlinna Theory: Positive logarithmic function Proximity function Counting function of zeros and poles Characteristic function Order and lower order of growth Nevanlinna’s First and Second Fundamental Theorems 00:37:57 – 00:41:19 📘 Discussion on *Recent Developments* in Nevanlinna Theory — differential–difference equations, transcendental entire functions, and modern results by researchers Harina P. Wagamore and Narin Kumar Bian. 00:40:40 – 00:42:31 🙏 *Conclusion by Dr. Nagarjun V* — expressing gratitude to Prajna Foundation and Dr. Mahesh Arvin for the opportunity. 00:42:01 – 00:44:26 🎤 *Vote of Thanks* by *Dr. Mahesh Arvin* — summary of talk highlights, appreciation for the speaker’s detailed explanation, and invitation for audience questions. 00:44:08 – 00:44:26 👋 Closing Note – thanking all participants and ending the session. Stay Connected 📺 YouTube: [@prajnafoundation2025](   / @prajnafoundation2025  ) 📷 Instagram: [@prajnafoundation2025](  / prajnafoundation2025  ) 🔖 🌐 #PrajnaFoundation #TalkOfTheWeek #ComplexAnalysis #Mathematics #NevanlinnaTheory #HigherEducation