# §6.16 Mathematical Applications

## §6.16(i) The Gibbs Phenomenon

Consider the Fourier series

 6.16.1 $\mathop{\sin\/}\nolimits x+\tfrac{1}{3}\mathop{\sin\/}\nolimits\!\left(3x% \right)+\tfrac{1}{5}\mathop{\sin\/}\nolimits\!\left(5x\right)+\dots=\begin{% cases}\frac{1}{4}\pi,&0 Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\sin\/}\nolimits\NVar{z}$: 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)

The $n$th partial sum is given by

 6.16.2 $S_{n}(x)=\sum_{k=0}^{n-1}\frac{\mathop{\sin\/}\nolimits\!\left((2k+1)x\right)}% {2k+1}=\frac{1}{2}\int_{0}^{x}\frac{\mathop{\sin\/}\nolimits\!\left(2nt\right)% }{\mathop{\sin\/}\nolimits t}\mathrm{d}t=\tfrac{1}{2}\mathop{\mathrm{Si}\/}% \nolimits\!\left(2nx\right)+R_{n}(x),$ Defines: $S_{n}(x)$: partial sum (locally) Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function, $\mathop{\mathrm{Si}\/}\nolimits\!\left(\NVar{z}\right)$: 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)

where

 6.16.3 $R_{n}(x)=\frac{1}{2}\int_{0}^{x}\left(\frac{1}{\mathop{\sin\/}\nolimits t}-% \frac{1}{t}\right)\mathop{\sin\/}\nolimits\!\left(2nt\right)\mathrm{d}t.$ Defines: $R_{n}(x)$: remainder term (locally) Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\sin\/}\nolimits\NVar{z}$: 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)

By integration by parts

 6.16.4 $R_{n}(x)=\mathop{O\/}\nolimits\!\left(n^{-1}\right),$ $n\to\infty$,

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 $\frac{1}{2}\mathop{\mathrm{Si}\/}\nolimits\!\left(x\right)$ 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.

## §6.16(ii) Number-Theoretic Significance of $\mathop{\mathrm{li}\/}\nolimits\!\left(x\right)$

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

 6.16.5 $\mathop{\mathrm{li}\/}\nolimits\!\left(x\right)-\pi(x)=\mathop{O\/}\nolimits\!% \left(\sqrt{x}\mathop{\ln\/}\nolimits x\right),$ $x\to\infty$,

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).