https://calculus.subwiki.org/w/index.php?action=history&feed=atom&title=Even_part
Even part - Revision history
2026-04-10T15:42:22Z
Revision history for this page on the wiki
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https://calculus.subwiki.org/w/index.php?title=Even_part&diff=363&oldid=prev
Vipul: /* Definition */
2011-09-22T20:05:13Z
<p><span dir="auto"><span class="autocomment">Definition</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:05, 22 September 2011</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\! f(x) = f_e(x) + f_o(x)</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>\! f(x) = f_e(x) + f_o(x)</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>with <math>f_e, f_o</math> both having the same domain as <math>f</math>, and with <math>f_e</math> an [[even function]] and <math>f_o</math> an [[odd function]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>with <math>f_e, f_o</math> both having the same domain as <math>f</math>, and with <math>f_e</math> an [[even function]] and <math>f_o</math> an [[odd function]]<ins style="font-weight: bold; text-decoration: none;">. The other part, <math>f_o</math>, is the [[odd part]] of <math>f</math></ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Particular cases==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Particular cases==</div></td></tr>
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Vipul
https://calculus.subwiki.org/w/index.php?title=Even_part&diff=98&oldid=prev
Vipul: /* Definition */
2011-08-28T12:43:57Z
<p><span dir="auto"><span class="autocomment">Definition</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:43, 28 August 2011</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l3">Line 3:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Suppose <math>f</math> is a function whose domain is a subset of the reals that is symmetric about 0, i.e., for every <math>x</math> in the domain of <math>f</math>, <math>-x</math> is also in the domain of <math>f</math>. Then, the '''even part''' of <math>f</math>, sometimes denoted <math>f_e</math> or <math>f_{\operatorname{even}}</math> is defined as a function with the same [[domain]], and with the definition:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Suppose <math>f</math> is a function whose domain is a subset of the reals that is symmetric about 0, i.e., for every <math>x</math> in the domain of <math>f</math>, <math>-x</math> is also in the domain of <math>f</math>. Then, the '''even part''' of <math>f</math>, sometimes denoted <math>f_e</math> or <math>f_{\operatorname{even}}</math> is defined as a function with the same [[domain]], and with the definition:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>f_e(x) := \frac{f(x) + f(-x)}{2}</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math><ins style="font-weight: bold; text-decoration: none;">\! </ins>f_e(x) := \frac{f(x) + f(-x)}{2}</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Equivalently, it is the only possible choice of [[defining ingredient::even function]] in a decomposition of <math>f</math> of the form:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Equivalently, it is the only possible choice of [[defining ingredient::even function]] in a decomposition of <math>f</math> of the form:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>f(x) = f_e(x) + f_o(x)</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math><ins style="font-weight: bold; text-decoration: none;">\! </ins>f(x) = f_e(x) + f_o(x)</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>with <math>f_e, f_o</math> both having the same domain as <math>f</math>, and with <math>f_e</math> an [[even function]] and <math>f_o</math> an [[odd function]].</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>with <math>f_e, f_o</math> both having the same domain as <math>f</math>, and with <math>f_e</math> an [[even function]] and <math>f_o</math> an [[odd function]].</div></td></tr>
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Vipul
https://calculus.subwiki.org/w/index.php?title=Even_part&diff=97&oldid=prev
Vipul: Created page with "==Definition== Suppose <math>f</math> is a function whose domain is a subset of the reals that is symmetric about 0, i.e., for every <math>x</math> in the domain of <math>f</mat..."
2011-08-28T12:42:57Z
<p>Created page with "==Definition== Suppose <math>f</math> is a function whose domain is a subset of the reals that is symmetric about 0, i.e., for every <math>x</math> in the domain of <math>f</mat..."</p>
<p><b>New page</b></p><div>==Definition==<br />
<br />
Suppose <math>f</math> is a function whose domain is a subset of the reals that is symmetric about 0, i.e., for every <math>x</math> in the domain of <math>f</math>, <math>-x</math> is also in the domain of <math>f</math>. Then, the '''even part''' of <math>f</math>, sometimes denoted <math>f_e</math> or <math>f_{\operatorname{even}}</math> is defined as a function with the same [[domain]], and with the definition:<br />
<br />
<math>f_e(x) := \frac{f(x) + f(-x)}{2}</math><br />
<br />
Equivalently, it is the only possible choice of [[defining ingredient::even function]] in a decomposition of <math>f</math> of the form:<br />
<br />
<math>f(x) = f_e(x) + f_o(x)</math><br />
<br />
with <math>f_e, f_o</math> both having the same domain as <math>f</math>, and with <math>f_e</math> an [[even function]] and <math>f_o</math> an [[odd function]].<br />
<br />
==Particular cases==<br />
<br />
{| class="sortable" border="1"<br />
! Function !! Domain !! Even part<br />
|-<br />
| polynomial || all of <math>\R</math> || the sum of the monomials of even degree in that polynomial<br />
|-<br />
| [[exponential function]] <math>e^x</math> || all of <math>\R</math> || [[hyperbolic cosine function]] <math>\cosh</math><br />
|}</div>
Vipul