15 terms

mechanic21PLUS

Handy reference: https://www.math24.net/even-odd-extensions/ (except they use pi instead of p for intervals)

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)

(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,...)

(because sin(0)=0,sin(pi)=0,sin(2pi)=0,...)