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Trigonometric Series

In: Introduction to Calculus and Analysis

Author

Listed:
  • Richard Courant

    (New York University, Courant Institute of Mathematical Sciences)

  • Fritz John

    (New York University, Courant Institute of Mathematical Sciences)

Abstract

The functions represented by power series, or as Lagrange called them, the “analytic functions,” play indeed a central role in analysis. But the class of analytic functions is too restricted in many instances. It was therefore an event of major importance for all of mathematics and for a great variety of applications when Fourier in his “Théorie analytique de la chaleur”1 observed and illustrated by many examples the fact that convergent trigonometric series of the form 1 $$f(x)=\frac{{{a_0}}}{2}+\sum\limits_{v = 1}^\infty {({a_v}\cos{\text{}}vx+{b_v}\sin{\text{}}vx)}$$ with constant coefficients a v , b v are capable of representing a wide class of “arbitrary” functions f(x), a class which includes essentially every function of specific interest, whether defined geometrically by mechanical means, or in any other way: even functions possessing jump discontinuities, or obeying different laws of formation in different intervals, can thus be expressed.

Suggested Citation

  • Richard Courant & Fritz John, 1989. "Trigonometric Series," Springer Books, in: Introduction to Calculus and Analysis, chapter 8, pages 571-632, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-8955-2_8
    DOI: 10.1007/978-1-4613-8955-2_8
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