§6.16 Mathematical Applications

§6.16(i) The Gibbs Phenomenon

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 ⓘ 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 $S_{n}(x)=\sum_{k=0}^{n-1}\frac{\sin\left((2k+1)x\right)}{2k+1}=\frac{1}{2}\int% _{0}^{x}\frac{\sin\left(2nt\right)}{\sin t}\mathrm{d}t=\tfrac{1}{2}\mathrm{Si}% \left(2nx\right)+R_{n}(x),$ ⓘ Defines: $S_{n}(x)$: partial sum (locally) Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\sin\NVar{z}$: sine function, $\mathrm{Si}\left(\NVar{z}\right)$: 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, $x=0$, or $-\pi; compare (6.2.14).

These limits are not approached uniformly, however. The first maximum of $\frac{1}{2}\mathrm{Si}\left(x\right)$ 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. Figure 6.16.1: Graph of Sn⁡(x), n=250, -0.1≤x≤0.1, illustrating the Gibbs phenomenon. Magnify

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

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 $\mathrm{li}\left(x\right)-\pi(x)=O\left(\sqrt{x}\ln x\right),$ $x\to\infty$,

where $\pi(x)$ 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). 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