Fourier Series - Calculation of Fourier Coefficients (continued)

In a Nut Shell: The Fourier Series expansion for the function, f(t), of period 2L is:

∞

f(t) = a_o /2 + ∑ a_n cos nπt/L + b_n sin nπt/L

n = 1

where the Fourier coefficients a_o , a_n , and b_n are determined by

L L L

a_o = (1/L) ∫ f(t) dt, a_n = (1/L) ∫ f(t) cos nπt/L dt b_n = (1/L) ∫ f(t) sin nπt/L

-L -L -L

Strategy: Determine if the function, f(t), is an even function, an odd function, or a function

that is neither even nor odd. Use this information to determine those coefficients that are

zero at the outset.

For “even” functions such as t², cos t, t sin t you only need to calculate the Fourier
coefficients a_o and a_n with all the b_n’s being zero. Note t sin t, is the product of an
odd function, t , with another odd function, sin t , which produces an "even" function where
f(t) = f(˗ t).

For “odd” functions such as t, sin t, t cos t you only need to calculate the Fourier
coefficients b_n’s with a_o and all the a_n being zero. Note t cos t is the product of the
odd function, t , with an even function, cos t , which produces an "odd" function
where f(-t) = ˗ f(t) .

If the function, f(t), is neither even nor odd, then all the Fourier Coefficients need to
be calculated.

In summary for f(t) and g(t):

f odd → a_n = 0 for all n ≥ 0 f even and g even → fg even

f even → b_n = 0 for all n ≥ 1 f odd and g odd → fg even

f odd and g even → fg odd or f even and g odd → fg odd

Note for an "even" function, f(t), the product of f(t) cos nπt/L is also "even" . Thus

L L

a_o = (1/L) ∫ f(t) dt, a_n = (1/L) ∫ f(t) cos nπt/L dt , b_n = 0

-L -L

L L

a_o = (2/L) ∫ f(t) dt, a_n = (2/L) ∫ f(t) cos nπt/L dt, b_n = 0

0 0

Click here for a concise summary.

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