Fourier Series - Summation to Integral

[erok81]

Homework Statement

My question involves the mid-point in deriving some of the equations to solve Laplace's equation in rectangular coordinates. The question may no make sense as it isn't problem specific. I can provide boundary values if necessary - just let me know.

Homework Equations

I've included a photo of how the example problem is broken up. For my question we'll choose subproblem #1.

$$u(x,y)=\sum^{\infty}_{n=1} A_{n} sin \frac{n \pi x}{a}sinh\frac{n\pi(b-y)}{a}$$

$$A_{n}= \frac{2}{a sinh \frac{n \pi b}{a}} \int^{a}_{0} f(x) sin \frac{n \pi x}{a} dx$$

The Attempt at a Solution

I don't understand how I get from the summation to the integral so I can solve for A_n. I see the pattern and transform the four summations in the example, but I'd really like to know the how/why it's done.

Let me know if I need to include anymore information as I don't have much regarding the actual problem.

 

[vela]

The non-zero boundary condition is

$$f(x) = u(x,0) = \sum^{\infty}_{m=1} \left(A_m\sinh\frac{m\pi b}{a}\right) \sin \frac{m \pi x}{a}$$

which is a Fourier series where the quantity in the parentheses is the m-th coefficient. Do you know how solve for the coefficients of a Fourier series?

 

[erok81]

I thought I knew how to find them.

When we were doing one dimensional waves I could find them no problem.

Where I'd have something like this.

$$\sum^{\infty}_{n=1} \left(B_{n} cos\lambda_{n}t+B^{*}_{n}sin\lambda_{n}t)sin(n \pi x)$$

To solve for B_n I'd take u(x,0)=f(x) and integrate.

$$\int^L_{-L} f(x) sin \frac{n \pi x}{L} dx$$

And then for the B^*_n I'd take u_t(x,0) = g(x) and do the same thing.

$$\int^L_{-L} g(x) sin \frac{n \pi x}{L} dx$$

Then the next chapter came up using 1D heat equation and the f(x) was given (and was no longer u(x,0) like the above example. Since it was always given I just threw it into it's appropriate location and solved away. Now, because I never knew why the f(x) became what it is, I am even more confused on how to find them.

So to answer you question, I guess I don't know how to find them.

 

[vela]

Suppose you have

$$f(x) = \sum_{m=1}^\infty b_m \sin \frac{m\pi x}{L}$$

Multiply both sides by sin (nπx/L) and integrate from 0 to L. Use the fact that

$$\int_0^L \sin \frac{m\pi x}{L}\sin \frac{n\pi x}{L}\,dx = \frac{L}{2}\delta_{mn}$$

where δ_mn is the Kronecker delta.