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 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. |
four cahâr (#) Fr.: quatre 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. |
four-dimensional operator âpârgar-e cahâr-vâmuni Fr.: opérateur à quatre dimensions An operator defined as: ▫ = (∂/∂x, ∂/∂y, ∂/∂z, 1/(jc∂/∂t). → four; → dimensional; → operator. |
Fourier analysis ânâlas-e Fourier 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. |
Fourier coefficient hamgar-e Fourier Fr.: coefficient de Fourier 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: → Fourier analysis; → series. |
Fourier integral dorostâl-e Fourier Fr.: intégrale de Fourier An integral used in the → Fourier transform. → Fourier analysis; → integral. |
Fourier series seri-ye Fourier 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: → Fourier analysis; → series. |
Fourier theorem farbin-e Fourier 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. → Fourier analysis; → theorem. |
Fourier transform tarâdis-e Fourier 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^{-iαu}du is called the Fourier transform of F(x) = (1/2π)∫ f(α)e^{iαx}dx, both integrals from -∞ to + ∞. → Fourier analysis; → transform. |
fourth contact parmâs-e cahârom 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. |