General Representation Formula for Harmonic Functions in the 2-D Case

[Muffins]

Homework Statement

Derive the Representation Formula for Harmonic Functions (i.e. $$\nabla^{2}u = 0$$)

Homework Equations

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The Attempt at a Solution

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Sorry about deferring to the pdf, but I already had everything typed up in LaTex previously, and I was having great difficulties getting the LaTex functionality of the in thread poster to work. I truly appreciate any suggestions or help that can be given

 

[mighty2000]

Thank you for your question regarding the representation formula for harmonic functions. I am happy to assist you in understanding this important concept.

Firstly, let's define what a harmonic function is. A harmonic function is a function that satisfies Laplace's equation, which is given by \nabla^{2}u = 0. This equation is also known as the Laplacian operator, and it represents the sum of the second partial derivatives of a function with respect to its variables.

Now, to derive the representation formula for harmonic functions, we will start by considering a solution to Laplace's equation in two dimensions, u(x,y). We can then write this solution in polar coordinates as u(r,\theta), where r represents the distance from the origin and \theta represents the angle with the positive x-axis. Using the chain rule, we can express the Laplacian operator in terms of polar coordinates as follows:

\nabla^{2}u = \frac{1}{r}\frac{\partial}{\partial r}\Big(r\frac{\partial u}{\partial r}\Big) + \frac{1}{r^{2}}\frac{\partial^{2}u}{\partial \theta^{2}}

Since u(r,\theta) is a solution to Laplace's equation, we know that \nabla^{2}u = 0. Therefore, we can rewrite the above equation as:

\frac{1}{r}\frac{\partial}{\partial r}\Big(r\frac{\partial u}{\partial r}\Big) + \frac{1}{r^{2}}\frac{\partial^{2}u}{\partial \theta^{2}} = 0

Now, let's focus on the first term of this equation. Using the product rule, we can expand it as follows:

\frac{1}{r}\frac{\partial}{\partial r}\Big(r\frac{\partial u}{\partial r}\Big) = \frac{\partial^{2}u}{\partial r^{2}} + \frac{1}{r}\frac{\partial u}{\partial r}

Substituting this into our original equation, we get:

\frac{\partial^{2}u}{\partial r^{2}} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^{2}}\frac{\partial