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6 Exponential, Logarithmic, Sine, and Cosine IntegralsApplications

§6.16 Mathematical Applications

  1. §6.16(i) The Gibbs Phenomenon
  2. §6.16(ii) Number-Theoretic Significance of li(x)

§6.16(i) The Gibbs Phenomenon

Consider the Fourier series

6.16.1 sinx+13sin(3x)+15sin(5x)+={14π,0<x<π,0,x=0,14π,π<x<0.

The nth partial sum is given by

6.16.2 Sn(x)=k=0n1sin((2k+1)x)2k+1=120xsin(2nt)sintdt=12Si(2nx)+Rn(x),


6.16.3 Rn(x)=120x(1sint1t)sin(2nt)dt.

By integration by parts

6.16.4 Rn(x)=O(n1),

uniformly for x[π,π]. Hence, if x is fixed and n, then Sn(x)14π, 0, or 14π according as 0<x<π, x=0, or π<x<0; compare (6.2.14).

These limits are not approached uniformly, however. The first maximum of 12Si(x) for positive x occurs at x=π and equals (1.1789)×14π; compare Figure 6.3.2. Hence if x=π/(2n) and n, then the limiting value of Sn(x) overshoots 14π by approximately 18%. Similarly if x=π/n, then the limiting value of Sn(x) undershoots 14π by approximately 10%, and so on. Compare Figure 6.16.1.

This nonuniformity of convergence is an illustration of the Gibbs phenomenon. It occurs with Fourier-series expansions of all piecewise continuous functions. See Carslaw (1930) for additional graphs and information.

See accompanying text
Figure 6.16.1: Graph of Sn(x), n=250, 0.1x0.1, illustrating the Gibbs phenomenon. Magnify

§6.16(ii) Number-Theoretic Significance of li(x)

If we assume Riemann’s hypothesis that all nonreal zeros of ζ(s) have real part of 1225.10(i)), then

6.16.5 li(x)π(x)=O(xlnx),

where π(x) is the number of primes less than or equal to x. Compare §27.12 and Figure 6.16.2. See also Bays and Hudson (2000).

See accompanying text
Figure 6.16.2: The logarithmic integral li(x), together with vertical bars indicating the value of π(x) for x=10,20,,1000. Magnify