Harmonic functions | Harmonic conjugate | Complex Analysis #3

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.

LINK TO COMPLEX ANALYSIS PLAYLIST
   • The Complete Guide to Complex Analysi...  

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IMPORTANT LINKS
Recap about the Cauchy Riemann Equations: https://goo.gl/QVM8pG
Cauchy Riemann Equations in polar form and cartesian form: https://goo.gl/Qxp3SS

CONCEPTS FROM THE VIDEO
► Harmonic Functions
Is a real-valued function u(x,y) with continuous second partial derivative which satisfies Laplace's equation.

► Harmonic Conjugate Functions
The harmonic conjugate to a given function u(x,y) is a function v(x,y) such that f(x,y) = u(x,y) + iv(x,y) is differentiable.

► Complex Differentiability
A function f(z) = u(x,y)+iv(x,y) is differentiable in a region R if and only if the following conditions are fulfilled in R:
1) du/dx, dv/dy, du/dy, dv/dx are continous
2) du/dx, dv/dy, du/dy, dv/dx satisfies the Cauchy Riemann Equations
The derivative is defined as f'(z) =du/dx + i*dv/dx=du/dy - i*dv/dy

► Analytic Continuation
It provides a way of extending the domain over which a complex function is defined. Let f_1 and f_2 be analytic functions which are defined on the domains d_1 and d_2, if f_1 = f_2 is true in the intersection of the domains then f_2 is called an analytic continuation of f_1 to d_2 and vice versa. This analytic continuation is unique if it exists.

TIMESTAMPS
Definition: Harmonic functions: 00:00 - 00:23
Theorem: Analytic function to Harmonic function: 00:23 - 00:48
Theorem: Analytic function to Harmonic function: 00:48 - 01:13
Definition: Harmonic functions: 01:13 - 01:35
Example: 01:35 - 01:59

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