Wikipédia

Dérét Fourier: Béda antarrépisi

Jelajahi riwayat secara interaktif
Éditan salajengna →
Konten dihapus Konten ditambahkan
Visual (béta)Téks wiki
Hadiyana (obrolan | kontribusi)
3.120 suntingan
Kaca anyar: Dina matematika, '''Dérét Fourier''' misah-misahkeun hiji fungsi périodik jadi sajumlah fungsi saderhana anu ngayunambing (osilasi), nyaéta [[gelombang sinus|sinus jeung kosinus]...
 
Hadiyana (obrolan | kontribusi)
3.120 suntingan
nuluykeun hanca
Baris ka-18:
{{cquote|<math>\varphi(y)=a\cos\frac{\pi y}{2}+a'\cos 3\frac{\pi y}{2}+a''\cos5\frac{\pi y}{2}+\cdots.</math>
 
MultiplyingKalikeun bothkadua sidessisi byjeung <math>\cos(2i+1)\frac{\pi y}{2}</math>, andsarta thentuluy integratingterapkeun fromoperasi integral ti <math>y=-1</math> tonepi ka <math>y=+1</math> yieldsmangka dihasilkeun:
 
<math>a_i=\int_{-1}^1\varphi(y)\cos(2i+1)\frac{\pi y}{2}\,dy.</math>
Baris ka-24:
|30px|30px|Joseph Fourier|Mémoire sur la propagation de la chaleur dans les corps solides, pp. 218--219.<ref>[http://gallica.bnf.fr/scripts/ConsultationTout.exe?O=03370&E=00000220&N=7 Gallica - Fourier, Jean-Baptiste-Joseph (1768-1830). Oeuvres de Fourier. 1888<!-- Bot generated title -->]</ref>}}
 
Dina sababaraha baris tulisan di luhur, Fourier sacara teu dihaja, ngalakukeun révolusi boh dina widang matematika boh dina widang fisika.
In these few lines, which are surprisingly close to the modern formalism used in Fourier series, Fourier unwittingly revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler, d'Alembert, Daniel Bernoulli and [[Carl Friedrich Gauss|Gauss]], Fourier believed that such trigonometric series could represent ''arbitrary'' functions. While this is not true, the attempts over many years to clarify this idea have led to important discoveries in the theories of [[convergence]], [[function space]]s, and [[harmonic analysis]].
 
===Lahirna analisis harmonik===
When Fourier submitted his paper in 1807, the committee (composed of no lesser mathematicians than [[Joseph Louis Lagrange|Lagrange]], [[Laplace]], [[Etienne-Louis Malus|Malus]] and [[Legendre]], among others) concluded: ''...the manner in which the author arrives at these equations is not exempt of difficulties and [...] his analysis to integrate them still leaves something to be desired on the score of generality and even rigour''.
Fourier mimitina ngadéfinisikeun dérét Fourier pikeun fungsi-fungsi nu boga harga ril sarta ngagunakeun fungsi-fungsi sinus jeung kosinus sabagé dasar pikeun misah-misahkeunana.
 
Saprak harita, kapanggih leuwih réa deui [[Daftar transformasi Fourier|transformasi Fourier]] nu bisa kadéfinisikeun, ngalébérkeun gagasan awal kana panerapan-panerapan séjénna. Widang nu néangan transformasi Fourier pikeun fungsi-fungsi éta disebut [[analisis harmonik]].
===The birth of harmonic analysis===
Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available at the time Fourier completed his original work. Fourier originally defined the Fourier series for real-valued functions of real arguments, and using the sine and cosine functions as the [[basis (linear algebra)|basis set]] for the decomposition.
 
Many other [[List of Fourier-related transforms|Fourier-related transforms]] have since been defined, extending the initial idea to other applications. This general area of inquiry is now sometimes called [[harmonic analysis]].
 
==Définisi==
 
In this section,Lamun ''fx''(''xt'') denotes angalambangkeun functionhiji offungsi theti realvariabel variablebébas ''xt''. Thismangka functionieu isfungsi usuallybiasana takendianggap to besabagé [[Periodic_function|periodic,fungsi périodik]] ofkalayan periodpérioda 2&pi;, whichdina iskalimah tolain saybisa thatdinyatakeun yén ''fx''(''xt''+2&pi;) = ''fx''(''xt''), forpikeun allsakabéh realangka numbersril ''xt''. WePikeun willnuliskeun showéta howfungsi to write such a function as an infinite sum,sabagé orpajumlahan [[seriesdérét (mathematicsmatematika)|seriesdérét]]. Wefungsi willsinusioda startanu bytanpa usingwates anréana, infiniteurang sumkudu ofngagunakeun pajumlahan fungsi-fungsi [[sinesinus]] andjeung [[cosinekosinus]] functionsanu oftanpa thewates dina interval [-&pi;,&pi;], assaperti anu dilakukeun ku Fourier did (seetingali thekutipan quotedi aboveluhur), and we will then discuss different formulations and generalizations.
 
===Rumus Fourier's formulapikeun forfungsi périodik 2&pi;-periodic functionsku usingngagunakeun fungsi-fungsi sinessinus andjeung cosineskosinus===
 
Pikeun hiji fungsi périodik 2&pi ''x''(t), angka-angka
For a 2&pi;-periodic function ''f''(x) the numbers
 
:<math>a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x(t) \cos(nxnt)\, dxdt</math>
 
jeung
and
 
:<math>b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x(t) \sin(nxnt)\, dxdt</math>
 
aredisebut called thekoéfisién Fourier coefficients oftina ''fx''. The [[infinitePajumlahan tanpa sumwates]]
 
:<math>f(x(t) = \frac{a_0}{2} +\sum_{n=1}^{\infty}[a_n \cos(nxnt) + b_n \sin(nxnt)]</math>
 
is themangrupakeun '''dérét Fourier series''' forpikeun ''fx'' on thedina interval [-&pi;,&pi;]. TheDérét Fourier series doeshenteu notkudu alwayskonvergén converge(ngurucut), so there may not be equality in the formula above. It is one of the main questions in [[Harmonic analysis]] to decide when equality holds. If a function is [[square-integrable]] on the interval [-&pi;,&pi;], then it can be represented in that interval by the previous formula.
 
=== Example: a simple Fourier series ===
Dicomot ti "https://su.wikipedia.org/wiki/Dérét_Fourier"