determination of Fourier coefficients
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
When , the equation (2) reads
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
to which we can include as a special case the equation (3).
By multiplying (1) by and integrating termwise, one obtains similarly
The equations (4) and (5) imply the formulas
for finding the values of the Fourier coefficients of .
|Title||determination of Fourier coefficients|
|Date of creation||2013-03-22 18:22:47|
|Last modified on||2013-03-22 18:22:47|
|Last modified by||pahio (2872)|
|Synonym||calculation of Fourier coefficients|