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| − | Formulas for the coefficients of a [[Fourier series|Fourier series]]. | + | Formulas for the coefficients of a [[Fourier series]]. |
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| − | ====Comments====
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| − | 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:
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| − | <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>
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| − | <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>
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| − | and
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| − | <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>
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| − | The formal series
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| − | <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>
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| − | and
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| − | <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>
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| − | 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:
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| − | <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>
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| − | 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" />.
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Latest revision as of 18:01, 2 May 2012
How to Cite This Entry:
Euler-Fourier formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler-Fourier_formulas&oldid=12665