- #1
fluidistic
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Homework Statement
I've reached a relation but then I need to obtain the coefficients ##A_m## and ##B_m##'s, those are the only unknowns.
Here's the expression: ##\sum _{m=0}^\infty a^m [A_m \cos (m \theta ) + B_m \sin (m \theta )]=T_0\sin ^3 \theta##.
Homework Equations
Fourier coefficients.
Result is probably ##B_1=\frac{3T_0}{4a}## and ##B_3=-\frac{T_0}{4a^3}##.
All the other coefficients being 0.
The Attempt at a Solution
So I am not really sure. I have something of the form ##\sum _{m=0}^\infty a^m [A_m \cos (m \theta ) + B_m \sin (m \theta )]=f(\theta) ##.
So in order to get let's say the ##A_m## first, I guess I must multiplicate both sides by ##cos (m \theta )## and integrate from ... 0 to ##2 \pi##?
I'd get ##\sum _{m=0}^\infty a^m A_m \int _0^{2\pi } \cos ^2 (m \theta ) d \theta + \sum _{m=0}^\infty a^m B_m \int _0^{2\pi } \sin (m \theta ) \cos (m\theta ) d \theta=\int _0 ^{2\pi } f(\theta ) d \theta##. Where an integral vanishes.
I'd be left with ##\sum _{m=0}^\infty a^m A_m \int _0^{2\pi } \cos ^2 (m \theta ) d \theta =\int _0 ^{2\pi } f(\theta ) d \theta##.
I don't really know how to evaluate those integrals nor how to isolate ##A_m##.