Harmonic Functions & Conjugates Explained | Complex Analysis #3

Описание: The definition of a Harmonic function, Harmonic conjugate function and how Analytic functions and Harmonic functions are related through some theorems. Examples for each concept are included.

Delve into the core definitions of harmonic functions and their conjugates! This video explores the fundamental relationships between analytic functions and harmonic functions in complex analysis, and walks through a step-by-step example to find a harmonic conjugate.

✨ In this lesson, you'll learn:
► The definition of a harmonic function (continuous second-order partial derivatives & Laplace's equation).
► The relationship between analytic functions and harmonic functions.
► How to find a harmonic conjugate v(x,y) for a given harmonic function u(x,y), leading to an analytic function F(z).

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📚 Resources & Playlists
Full Complex Analysis Course:    • Complex Analysis Explained (Full Course)  
Download the Lecture Notes: https://drive.google.com/file/d/0B0vT...

🔑 Key Concepts & Theorems
Harmonic Function
Harmonic Conjugate
Analytic Function
Laplace's Equation
Cauchy-Riemann Equations
Complex Differentiability

🔖 Chapters
00:00 Intro
00:21 Definition: Harmonic Function
00:48 Theorem: Analytic implies Harmonic
01:13 Theorem: Harmonic implies Analytic (Harmonic Conjugate exists)
01:36 Definition: Harmonic Conjugate
02:00 Example: Show u(x,y) is harmonic & find its conjugate v(x,y)
02:18 Showing u(x,y) is Harmonic
02:54 Finding the Harmonic Conjugate v(x,y)
04:45 Constructing the Analytic Function f(z)
05:15 Alternative Method: Analytic Continuation Trick
05:50 Outro

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