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Difference between revisions of "Euler-Fourier formulas"

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(comment duplicating material in Fouries Series deleted, as per discussion)
 
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Formulas for the coefficients of a [[Fourier series|Fourier series]].
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Formulas for the coefficients of a [[Fourier series]].
 
 
====Comments====
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036470/e0364701.png" /> is a function of period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036470/e0364702.png" /> defined on the real line and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036470/e0364703.png" />, then the formulas are:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036470/e0364704.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036470/e0364705.png" /></td> </tr></table>
 
 
 
and
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036470/e0364706.png" /></td> </tr></table>
 
 
 
The formal series
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036470/e0364707.png" /></td> </tr></table>
 
 
 
and
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036470/e0364708.png" /></td> </tr></table>
 
 
 
are both called the [[Fourier series|Fourier series]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036470/e0364709.png" />. They have the same partial sums:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036470/e03647010.png" /></td> </tr></table>
 
 
 
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036470/e03647011.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036470/e03647012.png" />. The coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036470/e03647013.png" /> is also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036470/e03647014.png" />.
 

Latest revision as of 20:01, 2 May 2012

Formulas for the coefficients of a Fourier series.

How to Cite This Entry:
Euler-Fourier formulas. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Euler-Fourier_formulas&oldid=25852