On Trigonometric Integrals

whose coefficients are expressed by the very well known formulas. The tests we have at our disposal for the convergence or summability of Fourier series are perfectly satisfactory for applications of Fourier series. In a number of problems we come across series which are of the form (1.1.1) without necessarily being Fourier series. Such series are the object of Riemann's theory of trigonometric series. This theory shows e.g. that if the series converges at every point of an interval (a, b) to a finite and integrable function f(x), then the series behaves essentially like a Fourier series. More precisely, if f*(x) is any integrable function of period 27r coinciding with f(x) in (a, b), and if

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