complex Fourier series
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 +π).
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.
Fr.: opérateur à quatre dimensions
An operator defined as: ▫ = (∂/∂x, ∂/∂y, ∂/∂z, 1/(jc∂/∂t).
Fr.: analyse de Fourier
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.
Fr.: coefficient de Fourier
One of the coefficients an or bn of cos (nx)
and sin (nx) respectively in the → Fourier series
representation of a function. They are expressed by:
Fr.: intégrale de Fourier
An integral used in the → Fourier transform.
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:
Fr.: théorème de Fourier
Any finite periodic motion may be analyzed into components, each of which is a simple harmonic motion of definite and determinable amplitudes and phase.
Fr.: transformée de Fourier
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)eiαxdx, both integrals from -∞ to + ∞.
Fr.: quatrième contact
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.