Harmonic functions | A Quick Proof | Complex Analysis #4

The Proof of why u(x,y) and v(x,y) are harmonic functions if f(z) = u(x,y) + iv(x,y) is an analytic function. This theorem is used in complex analysis and this video is a continuation of the video about Harmonic functions:    • Harmonic functions | Harmonic conjuga...  

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PROOF
First of the functions u(x,y) and v(x,y) have continous second order partial derivatives since f(z) = u(x,y) + iv(x,y) is analytic (this statement induces that f(z) is infinite differentiable, which makes u(x,y) and v(x,y) infinite differentiable aswell.).

Then the next step is to prove that the laplace´s equation is satisfied and this can be done with the help of the Cauchy Riemann equations and the fact that an analytic function most fulfill these equations. So we rewrite the laplace´s equation with the Cauchy Riemann equations and then we use the fact that u(x,y) and v(x,y) have continuous second order partial derivatives and therefore the mixed partial derivatives most be equal. Hence the laplace´s equationis satisfied.

Therefore the functions u(x,y) and v(x,y) are harmonic functions, since they fulfill the definition: continuous second order partial derivatives and satisfies the laplace´s equation.

TIMESTAMPS:
00:00 - 00:16 Intro
00:16 - 02:35 Proof
02:50 - 02:52 Outro

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