File:Gibbs phenomenon 50.svg

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$Picture of a function: sin ⁡ ( x ) + 1 3 sin ⁡ ( 3 x ) + 1 5 sin ⁡ ( 5 x ) + . . . + 1 49 sin ⁡ ( 49 x ) {\displaystyle \sin(x)+{\frac {1}{3}}\sin(3x)+{\frac {1}{5}}\sin(5x)+...+{\frac {1}{49}}\sin(49x)} as the 25-th approximation (the first twenty-five terms) of the wave function: f ( x ) = ∑ n = 1 ∞ 1 2 n − 1 sin ⁡ ( ( 2 n − 1 ) x ) {\displaystyle f(x)=\sum _{n=1}^{\infty }\,{\frac {1}{2n-1}}\sin((2n-1)x)} which is the Fourier series of the function: f ( x ) = { 0 x = 2 n π π / 4 2 n π < x < ( 2 n + 1 ) π 0 x = ( 2 n + 1 ) π − π / 4 ( 2 n + 1 ) π < x < 2 ( n + 1 ) π {\displaystyle f(x)={\begin{cases}0&x=2n\pi \\\pi /4&2n\pi <x<(2n+1)\pi \\0&x=(2n+1)\pi \\-\pi /4&(2n+1)\pi <x<2(n+1)\pi \\\end{cases}}}$

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Summary

DescriptionGibbs phenomenon 50.svg	Picture of a function: $\sin(x)+{\frac {1}{3}}\sin(3x)+{\frac {1}{5}}\sin(5x)+...+{\frac {1}{49}}\sin(49x)$ as the 25-th approximation (the first twenty-five terms) of the wave function: $f(x)=\sum _{n=1}^{\infty }\,{\frac {1}{2n-1}}\sin((2n-1)x)$ which is the Fourier series of the function: $f(x)={\begin{cases}0&x=2n\pi \\\pi /4&2n\pi <x<(2n+1)\pi \\0&x=(2n+1)\pi \\-\pi /4&(2n+1)\pi <x<2(n+1)\pi \\\end{cases}}$
Date	1 May 2010
Source	Own work, drawn with Inkscape's function plotter
Author	Johannes Rössel (talk)
Other versions	PNG version JPEG version

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File history

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	Date/Time	Dimensions	User	Comment
current	12:49, 1 May 2010	800 × 400 (7 KB)	Joey-das-WBF	line join
	12:09, 1 May 2010	800 × 400 (7 KB)	Joey-das-WBF	number of samples. Much smaller now
	12:07, 1 May 2010	800 × 400 (38 KB)	Joey-das-WBF	numeric precision; file is now much smaller
	12:03, 1 May 2010	800 × 400 (83 KB)	Joey-das-WBF	stroke width
	11:59, 1 May 2010	800 × 400 (83 KB)	Joey-das-WBF	interval
	11:57, 1 May 2010	800 × 400 (49 KB)	Joey-das-WBF	{{Valid SVG}} == {{int:filedesc}} == {{Information \|Description={{en\|Picture of a function: :<math>\sin(x)+\frac{1}{3}\sin(3x)+\frac{1}{5}\sin(5x)+...+\frac{1}{49}\sin(49x)</math> as the '''25-th''' approximation (the first twenty-five terms) of the wave