Loncat ke navigasi Loncat ke pencarian Dalam matematika , Deret Fourier merupakan penguraian fungsi periodik menjadi jumlahan fungsi-fungsi berosilasi, yaitu fungsi sinus dan kosinus, ataupun eksponensial kompleks. Studi deret Fourier merupakan cabang analisis Fourier. Deret Fourier diperkenalkan oleh Joseph Fourier untuk memecahkan masalah persamaan panas di lempeng logam. Persamaan panas merupakan persamaan diferensial parsial.
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Related transforms Linear operations performed in one domain time or frequency have corresponding operations in the other domain, which are sometimes easier to perform.
The operation of differentiation in the time domain corresponds to multiplication by the frequency, [remark 1] so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain see Convolution theorem. After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics.
Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function , of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution e.
The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer , where Gaussian functions appear as solutions of the heat equation. The Fourier transform can be formally defined as an improper Riemann integral , making it an integral transform , although this definition is not suitable for many applications requiring a more sophisticated integration theory.
This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics , where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued. The latter is routinely employed to handle periodic functions.
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