Basic General Formulas for Fourier Series PDF

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Basic General Formulas for Fourier Series Formula

Name, Comments ∞ X

f (x) = a0 +

an cos(wn x) + bn sin(wn x)

2 L-Periodic Definition

n=1

wn =

nπ L

a0 =

1 2L

Z L

f (x) dx

1 L

Z L

f (x) cos(wn x) dx

1 bn = L

Z L

f (x) sin(wn x) dx

an =

Euler formulae

−L

−L

f (x) =

−L ∞ X

cn ei wn x

Z L

f (x) e−i wn x dx

Complex form

n=−∞

cn =

1 2L

−L

f (x) = a0 +

∞ X

an cos(wn x)

Cosine series

n=1

a0 = an =

1 L

Z L

f (x) dx

2 L

Z L

f (x) cos(wn x) dx

f (x) =

0

0 ∞ X

bn sin(wn x)

Sine series

n=1

bn =

2 L

f (x) =

Z L

f (x) sin(wn x) dx

0 ∞ X

n=−∞

an =

(f, yn ) kyn k2

an yn (x)

Generalised series...