Suppose that the real function f may be presented as sum of the Fourier series:

f(x) = + ∑ m=0∞(a m cos mx + bm sin mx)

(1)

Therefore, f is periodic with period 2π. For expressing the Fourier coefficients a_m and b_m with the function itself, we first multiply the series (1) by cos nx (n ∈ ℤ) and integrate from −π to π. Supposing that we can integrate termwise, we may write

∫ −ππf(x) cos nxdx = ∫ −ππcos nxdx +∑ m=0∞.

(2)

When n = 0, the equation (2) reads

∫ −ππf(x) dx = ⋅ 2π = πa0,

(3)

since in the sum of the right hand side, only the first addend is distinct from zero.

When n 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 m≠n, the former equals zero because the antiderivative consists of sine terms which vanish at multiples of π; only in the case m = n we obtain from it a non-zero result π. Then (2) reads

∫ −ππf(x) cos nxdx = πa n

(4)

to which we can include as a special case the equation (3).

By multiplying (1) by sin nx and integrating termwise, one obtains similarly

∫ −ππf(x) sin nxdx = πb n.

(5)

The equations (4) and (5) imply the formulas

and

for finding the values of the Fourier coefficients of f.