سری ِ فوریهی ِ همتافت
seri-ye Fourier-ye hamtâft
Fr.: série de Fourier complexe
The complex notation for the → Fourier series of a function
f(x). Using → Euler's formulae, the function can be
written in cimplex form as
f(x) = Σ cn einx (summed from -∞ to ∞),
where the → Fourier coefficients are
cn = (1/2π)∫ f(x) e-inx dx (integral from -π to
→ complex; → Fourier series.
سری ِ فوریه
Fr.: séries Fourier
A mathematical tool used for decomposing a → periodic function
into an infinite sum of sine and cosine functions. The general form of the
Fourier series for a function f(x) with period 2π is:
(1/2) a0 + Σ (an cos (nx) +
bn sin (nx), summed from n = 1 to ∞,
where an and bn are the
→ Fourier coefficients, measuring
the strength of contribution from each harmonic.
The functions cos (nx) and sin (nx) can be used in this way because they
satisfy the → orthogonality conditions.
For the problem of convergence of the Fourier series see
→ Dirichlet conditions. The Fourier series
play a very important role in the study of periodic phenomena, because
they allow one to decompose a large number of complex problems into simpler ones.
The generalization of this method, called the → Fourier transform,
makes it possible to also decompose non-periodic functions into harmonic components.
See also → complex Fourier series,
→ Parseval's theorem.
→ Fourier analysis; → series.