The Fourier Transforms

  • Lokenath Debnath
  • Firdous A. Shah
Chapter
First Online:
Part of the Compact Textbooks in Mathematics book series (CTM)

Abstract

Historically, Joseph Fourier (1770–1830) first introduced the remarkable idea of expansion of a function in terms of trigonometric series without giving any attention to rigorous mathematical analysis. The integral formulas for the coefficients of the Fourier expansion were already known to Leonhard Euler (1707–1783) and others. In fact, Fourier developed his new idea for finding the solution of heat (or Fourier) equation in terms of Fourier series so that the Fourier series can be used as a practical tool for determining the Fourier series solution of partial differential equations under prescribed boundary conditions. Thus, the Fourier series of a function f(t) defined on the interval (−L, L) is given by

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Lokenath Debnath
    • 1
  • Firdous A. Shah
    • 2
  1. 1.School of Mathematical and Statistical SciencesUniversity of Texas – Rio Grande ValleyEdinburgUSA
  2. 2.Department of MathematicsUniversity of KashmirAnantnagIndia

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