6 Exponential, Logarithmic, Sine, and Cosine Integrals Applications 6.15 Sums 6.17 Physical Applications

§6.16 Mathematical Applications

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Permalink:: http://dlmf.nist.gov/6.16
See also:: Annotations for Ch.6

§6.16(i) The Gibbs Phenomenon

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Keywords:: Fourier-series expansions, Gibbs phenomenon, applications, maxima and minima, nonuniformity of convergence, piecewise continuous functions, sine integral, sine integrals
Notes:: See Temme (1996b, pp. 181–182: the numerical value $1.089490\dots$ on p. 182 should be replaced by $1.1789\dots$ ). Gibbs reported this phenomenon in a letter to Nature, 59 (1899, p. 606). Figure 6.16.1 was produced at NIST.
Referenced by:: §1.8(ii)
Permalink:: http://dlmf.nist.gov/6.16.i
See also:: Annotations for §6.16 and Ch.6

Consider the Fourier series

6.16.1		$\sin x+\tfrac{1}{3}\sin\left(3x\right)+\tfrac{1}{5}\sin\left(5x\right)+\dots=% \begin{cases}\frac{1}{4}\pi,&0<x<\pi,\\ 0,&x=0,\\ -\frac{1}{4}\pi,&-\pi<x<0.\end{cases}$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function and $x$ : real variable Permalink: http://dlmf.nist.gov/6.16.E1 Encodings: TeX, pMML, png See also: Annotations for §6.16(i), §6.16 and Ch.6

The $n$ th partial sum is given by

6.16.2		$S_{n}(x)=\sum_{k=0}^{n-1}\frac{\sin\left((2k+1)x\right)}{2k+1}=\frac{1}{2}\int% _{0}^{x}\frac{\sin\left(2nt\right)}{\sin t}\,\mathrm{d}t=\tfrac{1}{2}% \operatorname{Si}\left(2nx\right)+R_{n}(x),$
ⓘ Defines: $S_{n}(x)$ : partial sum (locally) Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral, $x$ : real variable, $n$ : nonnegative integer and $R_{n}(x)$ : remainder term Permalink: http://dlmf.nist.gov/6.16.E2 Encodings: TeX, pMML, png See also: Annotations for §6.16(i), §6.16 and Ch.6

where

6.16.3		$R_{n}(x)=\frac{1}{2}\int_{0}^{x}\left(\frac{1}{\sin t}-\frac{1}{t}\right)\sin% \left(2nt\right)\,\mathrm{d}t.$
ⓘ Defines: $R_{n}(x)$ : remainder term (locally) Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $x$ : real variable and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/6.16.E3 Encodings: TeX, pMML, png See also: Annotations for §6.16(i), §6.16 and Ch.6

By integration by parts

6.16.4		$R_{n}(x)=O\left(n^{-1}\right),$
$n\to\infty$ ,
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $x$ : real variable, $n$ : nonnegative integer and $R_{n}(x)$ : remainder term Permalink: http://dlmf.nist.gov/6.16.E4 Encodings: TeX, pMML, png See also: Annotations for §6.16(i), §6.16 and Ch.6

uniformly for $x\in[-\pi,\pi]$ . Hence, if $x$ is fixed and $n\to\infty$ , then $S_{n}(x)\to\frac{1}{4}\pi$ , $0$ , or $-\frac{1}{4}\pi$ according as $0<x<\pi$ , $x=0$ , or $-\pi<x<0$ ; compare (6.2.14).

These limits are not approached uniformly, however. The first maximum of $\frac{1}{2}\operatorname{Si}\left(x\right)$ for positive $x$ occurs at $x=\pi$ and equals $(1.1789\dots)\times\frac{1}{4}\pi$ ; compare Figure 6.3.2. Hence if $x=\pi/(2n)$ and $n\to\infty$ , then the limiting value of $S_{n}(x)$ overshoots $\frac{1}{4}\pi$ by approximately 18%. Similarly if $x=\pi/n$ , then the limiting value of $S_{n}(x)$ undershoots $\frac{1}{4}\pi$ 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 $S_{n}(x)$ , $n=250$ , $-0.1\leq x\leq 0.1$ , illustrating the Gibbs phenomenon. Magnify

§6.16(ii) Number-Theoretic Significance of $\operatorname{li}\left(x\right)$

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Keywords:: logarithmic integral, number-theoretic significance, prime numbers, relation to logarithmic integral
Notes:: Figure 6.16.2 was produced at NIST.
Permalink:: http://dlmf.nist.gov/6.16.ii
See also:: Annotations for §6.16 and Ch.6

If we assume Riemann’s hypothesis that all nonreal zeros of $\zeta\left(s\right)$ have real part of $\tfrac{1}{2}$ (§25.10(i)), then

6.16.5		$\operatorname{li}\left(x\right)-\pi(x)=O\left(\sqrt{x}\ln x\right),$
$x\to\infty$ ,
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\operatorname{li}\left(\NVar{x}\right)$ : logarithmic integral, $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real variable and $\pi(x)$ : number of primes $\leq x$ A&S Ref: 5.1.50 Permalink: http://dlmf.nist.gov/6.16.E5 Encodings: TeX, pMML, png See also: Annotations for §6.16(ii), §6.16 and Ch.6