An Etymological Dictionary of Astronomy and Astrophysics
English-French-Persian

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.

O.E. feower, from P.Gmc. *petwor- (cf. O.S. fiwar, Du. and Ger. vier, O.N. fjorir, Dan. fire, Sw. fyra), cognate with Pers. cahâr, as below, from PIE *qwetwor.

Cahâr, variant câr, from Mid.Pers. cahâr; Av. caθwarô, catur-; cf. Skt. catvārah; Gk. tessares; cognate with L. quattuor; E. four, as above.

An operator defined as: ▫ = (∂/∂x, ∂/∂y, ∂/∂z, 1/(jc∂/∂t).

→ four; → dimensional; → operator.

The process of decomposing any function of time or space into a sum of sinusoidal functions using the → Fourier series and → Fourier transforms. In other words, any data analysis procedure that describes or measures the fluctuations in a time series by comparing them with sinusoids. Fourier analysis is an essential component of much of modern applied and pure mathematics. It forms an exceptionally powerful analytical tool for solving various problems in many areas of mathematics, physics, engineering, biology, finance, etc. and has opened up new realms of knowledge.

After the French mathematician Baron Jean Baptiste Joseph Fourier (1768-1830), whose work had a tremendous impact on the physical applications of mathematics; → analysis.

One of the coefficients a_n or b_n of cos (nx) and sin (nx) respectively in the → Fourier series representation of a function. They are expressed by:
a_n = (1/π) ∫f(x) cos nx dx, for n≥ 0, summed over 0 to 2π
b_n = (1/π) ∫f(x) sin nx dx, for n≥ 1, summed over 0 to 2π.

→ Fourier analysis; → series.

An integral used in the → Fourier transform.

→ Fourier analysis; → integral.

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.

Any finite periodic motion may be analyzed into components, each of which is a simple harmonic motion of definite and determinable amplitudes and phase.

→ Fourier analysis; → theorem.

A powerful mathematical tool which is the generalization of the → Fourier series for the analysis of non-periodic functions. The Fourier transform transforms a function defined on physical space into a function defined on the space of frequencies, whose values quantify the "amount" of each periodic frequency contained in the original function. The inverse Fourier transform then reconstructs the original function from its transformed frequency components. The integral F(α) = ∫ f(u)e^-iαudu is called the Fourier transform of F(x) = (1/2π)∫ f(α)e^iαxdx, both integrals from -∞ to + ∞.

→ Fourier analysis; → transform.

The end of a solar eclipse marked by the disk of the Moon completely passing away from the disk of the Sun.

From M.E. fourthe, O.E. féowertha, from four, from O.E. feower, from P.Gmc. *petwor- (cf. Du. and Ger. vier, O.N. fjorir, Dan. fire, Sw. fyra), from PIE *qwetwor (cf. Mod.Pers. cahâr, Av. caθwar-, catur-, Skt. catvarah, Gk. tessares, L. quattuor) + -th a suffix used in the formation of ordinal numbers, from M.E. -the, -te, O.E. -tha, -the; cf. O.N. -thi, -di; L. -tus; Gk -tos; → contact.

Parmâs, → contact; cahârom cardinal form from cahâr "four," cognate with E. four, as above.