chapter  72
Fourier series over any range
Pages 7

L , then

du = 2π L

dx,

and the limits of integration are −L 2

to +L 2

instead of from −π to +π. Hence the Fourier

series expressed in terms of x is given by:

f (x) = a0 + ∞∑

[

an cos

( 2πnx

L

)

+ bn sin (

2πnx L

)]

where, in the range −L 2

to +L 2

:

and

a0 = 1L ∫ L

f (x) dx,

an = 2L ∫ L

f (x) cos (

2πnx L

)

dx

bn = 2L ∫ L

f (x) sin (

2πnx L

)

dx

The limits of integration may be replaced by any interval of length L, such as from 0 to L.