determination of Fourier coefficients
Suppose that the real function
may be presented as sum of the Fourier series:
 |
(1) |
Therefore,
is periodic with period
. For expressing the Fourier coefficients
and
with the function itself, we first multiply the series (1) by
(
) and integrate from
to
. Supposing that we can integrate termwise, we may write
 |
(2) |
When
, the equation (2) reads
 |
(3) |
since in the sum of the right hand side, only the first addend is distinct from zero.
When
is a positive integer, we use the product formulas
of the trigonometric identities, getting
The latter expression vanishes always, since the sine is an odd function. If
, the former equals zero because the antiderivative consists of sine terms which vanish at multiples of
; only in the case
we obtain from it a non-zero result
. Then (2) reads
 |
(4) |
to which we can include as a special case the equation (3).
By multiplying (1) by
and integrating termwise, one obtains similarly
 |
(5) |
The equations (4) and (5) imply the formulas
and
for finding the values of the Fourier coefficients of
.
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