# An Etymological Dictionary of Astronomy and AstrophysicsEnglish-French-Persian

## فرهنگ ریشه شناختی اخترشناسی-اخترفیزیک

### M. Heydari-Malayeri    -    Paris Observatory

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Number of Results: 10 Search : four
 complex Fourier series   سری ِ فوریه‌ی ِ همتافت   seri-ye Fourier-ye hamtâftFr.: 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). four   چهار   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   آپارگر ِ چهار-وامونی   âpârgar-e cahâr-vâmuniFr.: opérateur à quatre dimensions   An operator defined as: ▫ = (∂/∂x, ∂/∂y, ∂/∂z, 1/(jc∂/∂t).→ four; → dimensional; → operator. Fourier analysis   آنالس ِ فوریه   ânâlas-e FourierFr.: 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   همگر ِ فوریه   hamgar-e FourierFr.: 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: an = (1/π) ∫f(x) cos nx dx, for n≥ 0, summed over 0 to 2π bn = (1/π) ∫f(x) sin nx dx, for n≥ 1, summed over 0 to 2π.→ Fourier analysis; → series. Fourier integral   درستال ِ فوریه   dorostâl-e FourierFr.: intégrale de Fourier   An integral used in the → Fourier transform.→ Fourier analysis; → integral. Fourier series   سری ِ فوریه   seri-ye FourierFr.: 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. Fourier theorem   فربین ِ فوریه   farbin-e FourierFr.: 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   ترادیس ِ فوریه   tarâdis-e FourierFr.: 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 + ∞. fourth contact   پرماس ِ چهارم   parmâs-e cahâromFr.: 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.