An Etymological Dictionary of Astronomy and Astrophysics
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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 c_n e^inx (summed from -∞ to ∞), where the → Fourier coefficients are c_n = (1/2π)∫ f(x) e^-inx dx (integral from -π to +π).

→ complex; → Fourier series.

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 (a_n cos (nx) + b_n sin (nx), summed from n = 1 to ∞,
where a_n and b_n 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.