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[MA212] Unit 12: Fourier Series [Revised, Same as Before but Changed] (on the final exam!) (credit for set: jack_staveley) [term->def recommended]
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Handy reference: https://www.math24.net/even-odd-extensions/ (except they use pi instead of p for intervals)
Key Concepts:
Terms in this set (15)
Fourier Series on the interval (-p,p) (not an even or odd extension): f(x)=
Standard fourier series on interval (-p,p): a_0=
Standard fourier series on interval (-p,p): a_n=
Standard fourier series on interval (-p,p): b_n=
Even extension formula on the interval (0,p): f_e(x)=
Odd extension formula on the interval (0,p): f_o(x)=
NOTE! For odd extensions, there is no a_0 or extra addition term since you use b!!!
Even extension formula on the interval (0,p): a_0=
Even extension formula on the interval (0,p): a_n=
Odd extension formula on the interval (0,p): b_n=
If f is a piecewise differentiable function on (real interval) (-p,p) that is not continuous at some value of x (call it a), the fourier series of f(x) converges to:
If f is a piecewise differentiable function on (real interval) (-p,p) that is continuous at all values of x in this interval, the fourier series of f(x) converges to:
f(x) for all real values of x from (-p, p)
Odd Function (useful for odd extensions)
If f(x) = -f(-x)
Even Function (useful for even extensions)
If f(x) = f(-x)
Note (to simplify integrals from fourier series and extensions): cos(pi*k)= (where k is an integer)
(-1)^k
(because cos(0)=1,cos(pi)=-1,cos(2*pi)=1,..., so it's either -1 or 1 meaning you can get alternating terms depending on the value of k to simplify)
Note (to simplify integrals from fourier series and extensions): sin(pi*k)= (where k is an integer)
0
(because sin(0)=0,sin(pi)=0,sin(2pi)=0,...)
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