Fourier Series Flashcards
Inner products of sine and cosine =
0
If the inner product is zero a and b are said to be
orthogonal
Integration by parts
∫uv’ = u ∫v’ - ∫(u’( ∫v’))’
cos is an
even function
sin is an
odd function
for an even function
br = 0
for an odd function
ar = 0
basis vectors
they are orthogonal to each other
inner product of two sin functions
0 for r = p = 0
1/2 L for r = p > 0
0 for r ≠ p
inner product of two cosine functions
L for r = p = 0
1/2 L for r = p > 0
0 for r ≠ p
inner product of a cosine and sin function
= 0 for all r and p
Dirichlet conditions
(1) f(x) must be periodic
(2) f(x) must be single-valued and continuous
(3) f(x) must have only a finite number of extrema within one period
(4) the integral over one period must converge
value at discontinuity
f(t) = 0
Gibbs Phenomenon
add more terms -> constant value
( ∞ Σ n = - ∞) δ (ω-n) e^(iωt)
( ∞ Σ n = - ∞) e^(int)
