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. SUPPORT Consider subscribing and liking if you enjoyed this video or if it helped you understand the subject. It really helps me a lot. LINK TO CANVAS https://goo.gl/uvkVZU 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) which continuous second partial derivatives 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: 00:23 Theorem: Analytic function to Harmonic function: 00:48 Theorem: Analytic function to Harmonic function: 01:13 Definition: Harmonic functions: 01:35 Example: 01:59 SOCIAL ► Follow me on Youtube: http://bit.ly/1NQhPJ9 ► Follow me on Twitter: https://twitter.com/The_MathCoach ► Follow me on Facebook: https://www.facebook.com/TheMathCoach13/ ► Follow me on Google+: http://bit.ly/1DWqGsr