Talk:Fourier series: Difference between revisions

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href="/w/index.php?title=Talk:Fourier_series&action=edit&oldid=912494850" title="Talk:Fourier series">edit</a></span><span class="mw-diff-timestamp" data-timestamp="2019-08-26T00:23:11Z"></span></strong></div><div id="mw-diff-otitle2"><a href="/wiki/User:Lowercase_sigmabot_III" class="mw-userlink" title="User:Lowercase sigmabot III" data-mw-revid="912494850"><bdi>Lowercase sigmabot III</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:Lowercase_sigmabot_III" class="mw-redirect mw-usertoollinks-talk" title="User talk:Lowercase sigmabot III">talk</a> | <a href="/wiki/Special:Contributions/Lowercase_sigmabot_III" class="mw-usertoollinks-contribs" title="Special:Contributions/Lowercase sigmabot III">contribs</a>)</span><div class="mw-diff-usermetadata"><div class="mw-diff-userroles"><a href="/wiki/Wikipedia:Bots" title="Wikipedia:Bots">Bots</a>, <a href="/wiki/Wikipedia:Template_editor" title="Wikipedia:Template editor">Template editors</a></div><div class="mw-diff-usereditcount"><span>2,271,008</span> edits</div></div></div><div id="mw-diff-otitle3"><abbr class="minoredit" title="This is a minor edit">m</abbr> <span class="comment comment--without-parentheses">Archiving 2 discussion(s) to <a href="/wiki/Talk:Fourier_series/Archive_2" title="Talk:Fourier series/Archive 2">Talk:Fourier series/Archive 2</a>) (bot</span></div><div id="mw-diff-otitle5"></div><div id="mw-diff-otitle4"><a href="/w/index.php?title=Talk:Fourier_series&diff=prev&oldid=912494850" title="Talk:Fourier series" id="differences-prevlink">← Previous edit</a></div></td> <td colspan="2" class="diff-ntitle diff-side-added"><div id="mw-diff-ntitle1"><strong><a href="/w/index.php?title=Talk:Fourier_series&oldid=960025892" title="Talk:Fourier series">Revision as of 19:16, 31 May 2020</a> <span class="mw-diff-edit"><a href="/w/index.php?title=Talk:Fourier_series&action=edit&oldid=960025892" title="Talk:Fourier series">edit</a></span><span class="mw-diff-timestamp" data-timestamp="2020-05-31T19:16:49Z"></span> <span class="mw-diff-undo"><a href="/w/index.php?title=Talk:Fourier_series&action=edit&undoafter=912494850&undo=960025892" title=""Undo" reverts this edit and opens the edit form in preview mode. It allows adding a reason in the summary.">undo</a></span></strong></div><div id="mw-diff-ntitle2"><a href="/w/index.php?title=User:Thellamabotherer&action=edit&redlink=1" class="new mw-userlink" title="User:Thellamabotherer (page does not exist)" data-mw-revid="960025892"><bdi>Thellamabotherer</bdi></a> <span class="mw-usertoollinks">(<a href="/w/index.php?title=User_talk:Thellamabotherer&action=edit&redlink=1" class="new mw-usertoollinks-talk" title="User talk:Thellamabotherer (page does not exist)">talk</a> | <a href="/wiki/Special:Contributions/Thellamabotherer" class="mw-usertoollinks-contribs" title="Special:Contributions/Thellamabotherer">contribs</a>)</span><div class="mw-diff-usermetadata"><div class="mw-diff-userroles"></div><div class="mw-diff-usereditcount"><span>1</span> edit</div></div></div><div id="mw-diff-ntitle3"> <span class="comment comment--without-parentheses"><span class="autocomment"><a href="#Coefficient_of_a_0">→<bdi dir="ltr">Coefficient of a_0</bdi></a>: </span> new section</span></div><div id="mw-diff-ntitle5"></div><div id="mw-diff-ntitle4"><a href="/w/index.php?title=Talk:Fourier_series&diff=next&oldid=960025892" title="Talk:Fourier series" id="differences-nextlink">Next edit →</a></div></td> </tr><tr> <td colspan="2" class="diff-lineno">Line 111:</td> <td colspan="2" class="diff-lineno">Line 111:</td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><br /></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><br /></td> </tr> <tr> <td class="diff-marker"></td> <td class="diff-context diff-side-deleted"><div>The gif showing the synthesis is fabulous, but perhaps it should be stated explicitly that it is the SUM of the four sine waves that creates it. <small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/2001:8003:E448:D401:7D27:BA42:5CB1:E259|2001:8003:E448:D401:7D27:BA42:5CB1:E259]] ([[User talk:2001:8003:E448:D401:7D27:BA42:5CB1:E259#top|talk]]) 08:23, 25 August 2019 (UTC)</small> </div></td> <td class="diff-marker"></td> <td class="diff-context diff-side-added"><div>The gif showing the synthesis is fabulous, but perhaps it should be stated explicitly that it is the SUM of the four sine waves that creates it. <small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/2001:8003:E448:D401:7D27:BA42:5CB1:E259|2001:8003:E448:D401:7D27:BA42:5CB1:E259]] ([[User talk:2001:8003:E448:D401:7D27:BA42:5CB1:E259#top|talk]]) 08:23, 25 August 2019 (UTC)</small> </div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>== Coefficient of <math>a_0</math> ==</div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Sorry if I'm just being stupid here, which is probably the case. </div></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><br /></td> </tr> <tr> <td colspan="2" class="diff-empty diff-side-deleted"></td> <td class="diff-marker" data-marker="+"></td> <td class="diff-addedline diff-side-added"><div>Shouldn't the coefficent of <math>a_0</math> (underneath the first equation box in the definition section) be <math>\frac{2}{P}</math> rather than <math>\frac{1}{P}</math>. This feels like an easy mistake to make when <math>P</math> is normally going to be <math>2 \pi</math>.</div></td> </tr> </table><hr class='diff-hr' id='mw-oldid' /> <h2 class='diff-currentversion-title'>Revision as of 19:16, 31 May 2020</h2> <div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p><a href="/w/index.php?title=Template:Vital_article&action=edit&redlink=1" class="new" title="Template:Vital article (page does not exist)">Template:Vital article</a><style data-mw-deduplicate="TemplateStyles:r1237879475">.mw-parser-output .tmbox{margin:4px 0;border-collapse:collapse;border:1px solid #c0c090;background-color:#f8eaba;box-sizing:border-box}.mw-parser-output .tmbox.mbox-small{font-size:88%;line-height:1.25em}.mw-parser-output .tmbox-speedy{border:2px solid #b32424;background-color:#fee7e6}.mw-parser-output .tmbox-delete{border:2px solid #b32424}.mw-parser-output .tmbox-content{border:2px solid #f28500}.mw-parser-output .tmbox-style{border:2px solid 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Sections older than <b>90 days</b> may be automatically archived by <span class="nowraplinks"><a href="/wiki/User:Lowercase_sigmabot_III" title="User:Lowercase sigmabot III">Lowercase sigmabot III</a></span> when more than 5 sections are present.</td></tr> </tbody></table> <meta property="mw:PageProp/toc"/> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Eye-catching_GIF" data-mw-thread-id="h-Eye-catching_GIF-2014-03-07T13:48:00.000Z"><span data-mw-comment-start="" id="h-Eye-catching_GIF-2014-03-07T13:48:00.000Z"></span>Eye-catching GIF<span data-mw-comment-end="h-Eye-catching_GIF-2014-03-07T13:48:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <span data-mw-comment-start="" id="c-Bob_K-2014-03-07T13:48:00.000Z-Eye-catching_GIF"></span><figure typeof="mw:File/Thumb"><a href="/wiki/File:SquareWaveFourierArrows.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6b/SquareWaveFourierArrows.gif" decoding="async" width="300" height="90" class="mw-file-element" data-file-width="300" data-file-height="90"/></a><figcaption>A Fourier series enables any <a href="/wiki/Periodic_function" title="Periodic function">periodic motion</a>, such as the <a href="/wiki/Square_wave" title="Square wave">square wave</a> motion of the blue circle at the bottom, to be represented as the sum of many (often infinitely many) <a href="/wiki/Sine_wave" title="Sine wave">sinusoidally-oscillating</a> terms with different <a href="/wiki/Frequency" title="Frequency">frequencies</a>. The arrows show the six largest terms in the Fourier series, which sum to the purple circle.</figcaption></figure> <p>It's cool, but IMO it's sideways. Amplitude is customarily vertical. Partly for that reason, I think the only people who will actually comprehend it are those who have already internalized File:Fourier_Series.svg and are able to perform the mental (or laptop) rotation. For them, it is just a novelty. For others, it might serve the purpose of attracting attention, like a blinking light. But hopefully they won't get "stuck" there. My 2 cents. </p><p>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series#c-Bob_K-2014-03-07T13:48:00.000Z-Eye-catching_GIF" class="ext-discussiontools-init-timestamplink">13:48, 7 March 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2014-03-07T13:48:00.000Z-Eye-catching_GIF"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Bob_K-2014-03-07T13:48:00.000Z-Eye-catching_GIF"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Sbyrnes321-2014-03-07T16:53:00.000Z-Bob_K-2014-03-07T13:48:00.000Z"></span>It's not vertical because tall and narrow figures tend to make wikipedia page layout very difficult, compared to short and wide figures. (Or is there a trick I'm not aware of?)</dd> <dd>If you think the figure is doing more harm than good, you are entitled to delete it! --<a href="/wiki/User:Sbyrnes321" title="User:Sbyrnes321">Steve</a> (<a href="/wiki/User_talk:Sbyrnes321" title="User talk:Sbyrnes321">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series#c-Sbyrnes321-2014-03-07T16:53:00.000Z-Bob_K-2014-03-07T13:48:00.000Z" class="ext-discussiontools-init-timestamplink">16:53, 7 March 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Sbyrnes321-2014-03-07T16:53:00.000Z-Bob_K-2014-03-07T13:48:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Sbyrnes321-2014-03-07T16:53:00.000Z-Bob_K-2014-03-07T13:48:00.000Z"></span></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Sbyrnes321-2014-03-08T02:04:00.000Z-Bob_K-2014-03-07T13:48:00.000Z"></span>Incidentally, even disregarding page layout issues, I'm not convinced that vertical is really better. Fourier series the concept can appear in many superficially different settings: Vertical position as a function of horizontal position (your favorite), horizontal position as a function of time (my animation), vertical position as a function of time (your preference for my animation), color as a function of position (image processing), and on and on. Presenting the same concept in more than one superficial setting has a pedagogical value: It helps readers construct a more mature and refined conceptual understanding.</dd></dl> <dl><dd>(Yes, I acknowledge that this philosophy should not be taken to an extreme, where the settings are so bizarre that they are distracting and frustrating. But I think that in this particular situation, horizontal is OK.)</dd></dl> <dl><dd>Also, if it stays at the top of the article, it may be the first thing that people ever see about Fourier series, so they would not necessarily have the same preconception that you have, i.e. "x(t) is weird and y(t) is normal". --<a href="/wiki/User:Sbyrnes321" title="User:Sbyrnes321">Steve</a> (<a href="/wiki/User_talk:Sbyrnes321" title="User talk:Sbyrnes321">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series#c-Sbyrnes321-2014-03-08T02:04:00.000Z-Bob_K-2014-03-07T13:48:00.000Z" class="ext-discussiontools-init-timestamplink">02:04, 8 March 2014 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Sbyrnes321-2014-03-08T02:04:00.000Z-Bob_K-2014-03-07T13:48:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Sbyrnes321-2014-03-08T02:04:00.000Z-Bob_K-2014-03-07T13:48:00.000Z"></span></dd></dl> <p><span data-mw-comment-start="" id="c-2601:703:0:851E:91E5:CA2D:218:296C-2016-02-25T16:27:00.000Z-Eye-catching_GIF"></span>I should like to differ on the description of the Fourier series. First this definition of a Fourier series is applicable only to scientists and engineers. Hence is not really mathematics but engineering. In mathematics a Fourier series is defined within any inner product space within a maximal orthonormal set within that inner product space. The engineering version set out in this article is valid only for the inner product space L2[a,b]. <small class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/2601:703:0:851E:91E5:CA2D:218:296C" title="Special:Contributions/2601:703:0:851E:91E5:CA2D:218:296C">2601:703:0:851E:91E5:CA2D:218:296C</a> (<a href="/w/index.php?title=User_talk:2601:703:0:851E:91E5:CA2D:218:296C&action=edit&redlink=1" class="new" title="User talk:2601:703:0:851E:91E5:CA2D:218:296C (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series#c-2601:703:0:851E:91E5:CA2D:218:296C-2016-02-25T16:27:00.000Z-Eye-catching_GIF" class="ext-discussiontools-init-timestamplink">16:27, 25 February 2016 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-2601:703:0:851E:91E5:CA2D:218:296C-2016-02-25T16:27:00.000Z-Eye-catching_GIF"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-2601:703:0:851E:91E5:CA2D:218:296C-2016-02-25T16:27:00.000Z-Eye-catching_GIF"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="there_is_an_asterisk_(*)._What_does_it_mean?" data-mw-thread-id="h-there_is_an_asterisk_(*)._What_does_it_mean?-2016-06-16T08:59:00.000Z"><span id="there_is_an_asterisk_.28.2A.29._What_does_it_mean.3F"></span><span data-mw-comment-start="" id="h-there_is_an_asterisk_(*)._What_does_it_mean?-2016-06-16T08:59:00.000Z"></span>there is an asterisk (*). What does it mean?<span data-mw-comment-end="h-there_is_an_asterisk_(*)._What_does_it_mean?-2016-06-16T08:59:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-88.188.110.51-2016-06-16T08:59:00.000Z-there_is_an_asterisk_(*)._What_does_it_mean?"></span>Dear Author, please accept my many thanks for your helpful article, and permit me a question: in "Definition" above "we can also write the function in these equivalent forms:" at the end of formula, there is an asterisk (*). What does it mean? Regards, Georges Theodosiou, <a href="/wiki/Special:Contributions/88.188.110.51" title="Special:Contributions/88.188.110.51">88.188.110.51</a> (<a href="/wiki/User_talk:88.188.110.51" title="User talk:88.188.110.51">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series#c-88.188.110.51-2016-06-16T08:59:00.000Z-there_is_an_asterisk_(*)._What_does_it_mean?" class="ext-discussiontools-init-timestamplink">08:59, 16 June 2016 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-88.188.110.51-2016-06-16T08:59:00.000Z-there_is_an_asterisk_(*)._What_does_it_mean?"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-88.188.110.51-2016-06-16T08:59:00.000Z-there_is_an_asterisk_(*)._What_does_it_mean?"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2016-06-16T12:56:00.000Z-88.188.110.51-2016-06-16T08:59:00.000Z"></span>It is a common notation for the <a href="/wiki/Complex_conjugate#Notation" title="Complex conjugate">Complex_conjugate#Notation</a> operation. --<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series#c-Bob_K-2016-06-16T12:56:00.000Z-88.188.110.51-2016-06-16T08:59:00.000Z" class="ext-discussiontools-init-timestamplink">12:56, 16 June 2016 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2016-06-16T12:56:00.000Z-88.188.110.51-2016-06-16T08:59:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Bob_K-2016-06-16T12:56:00.000Z-88.188.110.51-2016-06-16T08:59:00.000Z"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-88.188.110.51-2016-06-16T13:48:00.000Z-Bob_K-2016-06-16T12:56:00.000Z"></span>Mr Author, please let me express you my sincere thanks for your helpful answer. Georges Theodosiou <a href="/wiki/Special:Contributions/88.188.110.51" title="Special:Contributions/88.188.110.51">88.188.110.51</a> (<a href="/wiki/User_talk:88.188.110.51" title="User talk:88.188.110.51">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series#c-88.188.110.51-2016-06-16T13:48:00.000Z-Bob_K-2016-06-16T12:56:00.000Z" class="ext-discussiontools-init-timestamplink">13:48, 16 June 2016 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-88.188.110.51-2016-06-16T13:48:00.000Z-Bob_K-2016-06-16T12:56:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-88.188.110.51-2016-06-16T13:48:00.000Z-Bob_K-2016-06-16T12:56:00.000Z"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="The_first_few_paragraphs_should_be_moved" data-mw-thread-id="h-The_first_few_paragraphs_should_be_moved"><span data-mw-comment-start="" id="h-The_first_few_paragraphs_should_be_moved"></span>The first few paragraphs should be moved<span data-mw-comment-end="h-The_first_few_paragraphs_should_be_moved"></span></h2></div> <p>From the engineering and the maths perspective, those 3 paragraphs aren't helpful for the reader where they are placed : </p> <pre> 1 History 2 Definition 2.1 Complex-valued functions </pre> <p>We need to start with the definition of the series and the coefficients, the visual gif plus the problem of the convergence, the whole in less than 10 lines. Take a look at <a href="/wiki/Convergence_of_Fourier_series" title="Convergence of Fourier series">Convergence of Fourier series</a>. Only after that, we can go into disgression, the intuitive ideas, the theorems, the history, the real vs complex notation. Assuming the reader doesn't know what is a series and a complex number is ridiculous, as well as beginning the article with the partial sums. <a href="/w/index.php?title=User:Reuns&action=edit&redlink=1" class="new" title="User:Reuns (page does not exist)">Reuns</a> (<a href="/wiki/User_talk:Reuns" title="User talk:Reuns">talk</a>) </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Fourier_Series_definition_and_other_problems" data-mw-thread-id="h-Fourier_Series_definition_and_other_problems-2017-10-17T20:33:00.000Z"><span data-mw-comment-start="" id="h-Fourier_Series_definition_and_other_problems-2017-10-17T20:33:00.000Z"></span>Fourier Series definition and other problems<span data-mw-comment-end="h-Fourier_Series_definition_and_other_problems-2017-10-17T20:33:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bob_K-2017-10-17T20:33:00.000Z-Fourier_Series_definition_and_other_problems"></span>There is a problem with the first equation on the page. The Fourier Series is defined as a series with amplitude and phase angle for each harmonic then the cosine function should be used. This is consistent with the idea of "phasor". The consistent Fourier Series definition follows as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{N}(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{N}A_{n}\times \cos \left({\tfrac {2\pi nx}{P}}+\phi _{n}\right),\quad {\text{for integer}}\ N\ \geq \ 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>×</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>for integer</mtext> </mrow> <mtext> </mtext> <mi>N</mi> <mtext> </mtext> <mo>≥</mo> <mtext> </mtext> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{N}(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{N}A_{n}\times \cos \left({\tfrac {2\pi nx}{P}}+\phi _{n}\right),\quad {\text{for integer}}\ N\ \geq \ 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d40cec2b1f57f5e1063ec1d3a2b37a58d2102323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:63.314ex; height:7.343ex;" alt="{\displaystyle s_{N}(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{N}A_{n}\times \cos \left({\tfrac {2\pi nx}{P}}+\phi _{n}\right),\quad {\text{for integer}}\ N\ \geq \ 1.}"/></span></dd></dl> <p>Note, I have also used the multiplication symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"/></span> instead of the dot product symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }"/></span>. From an aesthetic perspective, I would even prefer to remove the multiplication symbol completely. </p><p>Another problem is in the definition of the equivalent forms. The identity only holds if the sine part is complex, as shown in the following equation: </p> <table class="wikitable" style="text-align:left"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s_{N}(x)&=\overbrace {a_{0}} ^{A_{0}}/2+\sum _{n=1}^{N}\left(\overbrace {a_{n}} ^{A_{n}\sin(\phi _{n})}\cos \left({\tfrac {2\pi nx}{P}}\right)+\overbrace {b_{n}} ^{A_{n}\cos(\phi _{n})}i\,\sin \left({\tfrac {2\pi nx}{P}}\right)\right)\\&=\sum _{n=-N}^{N}c_{n}\times e^{i{\tfrac {2\pi nx}{P}}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mover> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mover> <mi>i</mi> <mspace width="thinmathspace"></mspace> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>×</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{N}(x)&=\overbrace {a_{0}} ^{A_{0}}/2+\sum _{n=1}^{N}\left(\overbrace {a_{n}} ^{A_{n}\sin(\phi _{n})}\cos \left({\tfrac {2\pi nx}{P}}\right)+\overbrace {b_{n}} ^{A_{n}\cos(\phi _{n})}i\,\sin \left({\tfrac {2\pi nx}{P}}\right)\right)\\&=\sum _{n=-N}^{N}c_{n}\times e^{i{\tfrac {2\pi nx}{P}}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1635a515222a732d0f5f178e63df2bb37faad838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:67.807ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}s_{N}(x)&=\overbrace {a_{0}} ^{A_{0}}/2+\sum _{n=1}^{N}\left(\overbrace {a_{n}} ^{A_{n}\sin(\phi _{n})}\cos \left({\tfrac {2\pi nx}{P}}\right)+\overbrace {b_{n}} ^{A_{n}\cos(\phi _{n})}i\,\sin \left({\tfrac {2\pi nx}{P}}\right)\right)\\&=\sum _{n=-N}^{N}c_{n}\times e^{i{\tfrac {2\pi nx}{P}}},\end{aligned}}}"/></span> </td></tr></tbody></table> <p>Please discuss and comment. </p> <dl><dd>The generalization to complex-valued s(x) happens in section <a href="/wiki/Fourier_series#Complex-valued_functions" title="Fourier series">Fourier_series#Complex-valued_functions</a>.</dd></dl> <dl><dd>As clearly stated in the article, the identity you don't understand is just basic trigonometry:</dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \left({\tfrac {2\pi nx}{P}}+\phi _{n}\right)\equiv \sin(\phi _{n})\cos \left({\tfrac {2\pi nx}{P}}\right)+\cos(\phi _{n})\sin \left({\tfrac {2\pi nx}{P}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>≡</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \left({\tfrac {2\pi nx}{P}}+\phi _{n}\right)\equiv \sin(\phi _{n})\cos \left({\tfrac {2\pi nx}{P}}\right)+\cos(\phi _{n})\sin \left({\tfrac {2\pi nx}{P}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3749dbab6378387aa48f5155fa5c7f7d51a35bcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.325ex; height:4.843ex;" alt="{\displaystyle \sin \left({\tfrac {2\pi nx}{P}}+\phi _{n}\right)\equiv \sin(\phi _{n})\cos \left({\tfrac {2\pi nx}{P}}\right)+\cos(\phi _{n})\sin \left({\tfrac {2\pi nx}{P}}\right).}"/></span></dd></dl></dd></dl> <dl><dd>Therefore the formula:</dd></dl> <dl><dd><table class="wikitable" style="text-align:left"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s_{N}(x)&=\overbrace {a_{0}} ^{A_{0}}/2+\sum _{n=1}^{N}\left(\overbrace {a_{n}} ^{A_{n}\sin(\phi _{n})}\cos \left({\tfrac {2\pi nx}{P}}\right)+\overbrace {b_{n}} ^{A_{n}\cos(\phi _{n})}\sin \left({\tfrac {2\pi nx}{P}}\right)\right)\\&=\sum _{n=-N}^{N}c_{n}\cdot e^{i{\tfrac {2\pi nx}{P}}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mover> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mover> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mover> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{N}(x)&=\overbrace {a_{0}} ^{A_{0}}/2+\sum _{n=1}^{N}\left(\overbrace {a_{n}} ^{A_{n}\sin(\phi _{n})}\cos \left({\tfrac {2\pi nx}{P}}\right)+\overbrace {b_{n}} ^{A_{n}\cos(\phi _{n})}\sin \left({\tfrac {2\pi nx}{P}}\right)\right)\\&=\sum _{n=-N}^{N}c_{n}\cdot e^{i{\tfrac {2\pi nx}{P}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2773b2d39ad9bf5b197a2b326ac5edda98146ed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:66.617ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}s_{N}(x)&=\overbrace {a_{0}} ^{A_{0}}/2+\sum _{n=1}^{N}\left(\overbrace {a_{n}} ^{A_{n}\sin(\phi _{n})}\cos \left({\tfrac {2\pi nx}{P}}\right)+\overbrace {b_{n}} ^{A_{n}\cos(\phi _{n})}\sin \left({\tfrac {2\pi nx}{P}}\right)\right)\\&=\sum _{n=-N}^{N}c_{n}\cdot e^{i{\tfrac {2\pi nx}{P}}}\end{aligned}}}"/></span> </td></tr></tbody></table></dd></dl> <dl><dd>is intentionally real-valued. And in case you missed it<b>:</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}\cdot e^{i{\tfrac {2\pi nx}{P}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}\cdot e^{i{\tfrac {2\pi nx}{P}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d16bbb7a335b6e2d83bbd05056c6f4a7f0784e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.314ex; height:4.343ex;" alt="{\displaystyle c_{n}\cdot e^{i{\tfrac {2\pi nx}{P}}}}"/></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{-n}\cdot e^{i{\tfrac {2\pi (-n)x}{P}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> <mi>P</mi> </mfrac> </mstyle> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{-n}\cdot e^{i{\tfrac {2\pi (-n)x}{P}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e2e075b2bdad9798f726a5eeacb10929f90b7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.15ex; height:4.843ex;" alt="{\displaystyle c_{-n}\cdot e^{i{\tfrac {2\pi (-n)x}{P}}}}"/></span> are complex conjugates. Their sum is real.</dd></dl> <dl><dd>The • operator is commonly used to represent the product of 2 operands, which technically <u>is</u> a one-dimensional <u>dot product</u>. So, at worst, you can call it overkill, if you insist on thinking in those terms. But I'm confident that most readers just see it as an alternative to ×, just as / is an alternative to ÷.</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series#c-Bob_K-2017-10-17T20:33:00.000Z-Fourier_Series_definition_and_other_problems" class="ext-discussiontools-init-timestamplink">20:33, 17 October 2017 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2017-10-17T20:33:00.000Z-Fourier_Series_definition_and_other_problems"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Bob_K-2017-10-17T20:33:00.000Z-Fourier_Series_definition_and_other_problems"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Dicklyon-2019-04-25T05:24:00.000Z-Bob_K-2017-10-17T20:33:00.000Z"></span>I think you're both right. That is, the derivation seems correct, but starting with the sine instead of the cosine for defining the sinusoid with a phase angle seems a bit unconventional, and makes it a little messier, like needing to divide by i to get c_n. <a href="/wiki/User:Dicklyon" title="User:Dicklyon">Dicklyon</a> (<a href="/wiki/User_talk:Dicklyon" title="User talk:Dicklyon">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series#c-Dicklyon-2019-04-25T05:24:00.000Z-Bob_K-2017-10-17T20:33:00.000Z" class="ext-discussiontools-init-timestamplink">05:24, 25 April 2019 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dicklyon-2019-04-25T05:24:00.000Z-Bob_K-2017-10-17T20:33:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Dicklyon-2019-04-25T05:24:00.000Z-Bob_K-2017-10-17T20:33:00.000Z"></span></dd></dl></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Bob_K-2019-04-26T23:33:00.000Z-Bob_K-2017-10-17T20:33:00.000Z"></span>"Conventional" is the sine-cosine form<b>:</b></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(2\pi \ n\ x)+b_{n}\sin(2\pi \ n\ x)].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mtext> </mtext> <mi>n</mi> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mtext> </mtext> <mi>n</mi> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(2\pi \ n\ x)+b_{n}\sin(2\pi \ n\ x)].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5974c3d6e83eb3fd04b7d61e3871f32113cc23a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.043ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(2\pi \ n\ x)+b_{n}\sin(2\pi \ n\ x)].}"/></span></dd></dl></dd></dl> <dl><dd>Unconventional would be<b>:</b></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(2\pi \ n\ x)-b_{n}\sin(2\pi \ n\ x)].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mtext> </mtext> <mi>n</mi> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mtext> </mtext> <mi>n</mi> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(2\pi \ n\ x)-b_{n}\sin(2\pi \ n\ x)].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20e7032d931bc1fa7769cd637eb929335ed4482a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.043ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(2\pi \ n\ x)-b_{n}\sin(2\pi \ n\ x)].}"/></span></dd></dl></dd></dl> <dl><dd>But the former suggests<b>:</b></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(2\pi nx+\varphi _{n})\equiv \sin(\varphi _{n})\cos(2\pi nx)+\cos(\varphi _{n})\sin(2\pi nx),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo>+</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≡</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(2\pi nx+\varphi _{n})\equiv \sin(\varphi _{n})\cos(2\pi nx)+\cos(\varphi _{n})\sin(2\pi nx),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2d58ebdd83e668555a3f4db4171af8bfc3bb116" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.908ex; height:2.843ex;" alt="{\displaystyle \sin(2\pi nx+\varphi _{n})\equiv \sin(\varphi _{n})\cos(2\pi nx)+\cos(\varphi _{n})\sin(2\pi nx),}"/></span></dd></dl></dd></dl> <dl><dd>because of consistently "+" signs. For sign consistency, the latter suggests<b>:</b></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(2\pi nx+\varphi _{n})\equiv \cos(\varphi _{n})\cos(2\pi nx)-\sin(\varphi _{n})\sin(2\pi nx),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo>+</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≡</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(2\pi nx+\varphi _{n})\equiv \cos(\varphi _{n})\cos(2\pi nx)-\sin(\varphi _{n})\sin(2\pi nx),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d02ef7b262b2497254a8a5fff9cd645878bd82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.164ex; height:2.843ex;" alt="{\displaystyle \cos(2\pi nx+\varphi _{n})\equiv \cos(\varphi _{n})\cos(2\pi nx)-\sin(\varphi _{n})\sin(2\pi nx),}"/></span></dd></dl></dd></dl> <dl><dd>which is also OK with me.</dd> <dd>--<a href="/wiki/User:Bob_K" title="User:Bob K">Bob K</a> (<a href="/wiki/User_talk:Bob_K" title="User talk:Bob K">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series#c-Bob_K-2019-04-26T23:33:00.000Z-Bob_K-2017-10-17T20:33:00.000Z" class="ext-discussiontools-init-timestamplink">23:33, 26 April 2019 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bob_K-2019-04-26T23:33:00.000Z-Bob_K-2017-10-17T20:33:00.000Z"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Bob_K-2019-04-26T23:33:00.000Z-Bob_K-2017-10-17T20:33:00.000Z"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Label_for_the_cool_gif_is_unclear." data-mw-thread-id="h-Label_for_the_cool_gif_is_unclear.-2019-08-25T08:23:00.000Z"><span data-mw-comment-start="" id="h-Label_for_the_cool_gif_is_unclear.-2019-08-25T08:23:00.000Z"></span>Label for the cool gif is unclear.<span data-mw-comment-end="h-Label_for_the_cool_gif_is_unclear.-2019-08-25T08:23:00.000Z"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-2001:8003:E448:D401:7D27:BA42:5CB1:E259-2019-08-25T08:23:00.000Z-Label_for_the_cool_gif_is_unclear."></span>The gif showing the synthesis is fabulous, but perhaps it should be stated explicitly that it is the SUM of the four sine waves that creates it. <small class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/Special:Contributions/2001:8003:E448:D401:7D27:BA42:5CB1:E259" title="Special:Contributions/2001:8003:E448:D401:7D27:BA42:5CB1:E259">2001:8003:E448:D401:7D27:BA42:5CB1:E259</a> (<a href="/w/index.php?title=User_talk:2001:8003:E448:D401:7D27:BA42:5CB1:E259&action=edit&redlink=1" class="new" title="User talk:2001:8003:E448:D401:7D27:BA42:5CB1:E259 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:Fourier_series#c-2001:8003:E448:D401:7D27:BA42:5CB1:E259-2019-08-25T08:23:00.000Z-Label_for_the_cool_gif_is_unclear." class="ext-discussiontools-init-timestamplink">08:23, 25 August 2019 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-2001:8003:E448:D401:7D27:BA42:5CB1:E259-2019-08-25T08:23:00.000Z-Label_for_the_cool_gif_is_unclear."><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-2001:8003:E448:D401:7D27:BA42:5CB1:E259-2019-08-25T08:23:00.000Z-Label_for_the_cool_gif_is_unclear."></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Coefficient_of_'"`UNIQ--postMath-0000000D-QINU`"'" data-mw-thread-id="h-Coefficient_of_'"`UNIQ--postMath-0000000D-QINU`"'"><span id="Coefficient_of_.7F.27.22.60UNIQ--postMath-0000000D-QINU.60.22.27.7F"></span><span data-mw-comment-start="" id="h-Coefficient_of_'"`UNIQ--postMath-0000000D-QINU`"'"></span>Coefficient of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"/></span><span data-mw-comment-end="h-Coefficient_of_'"`UNIQ--postMath-0000000D-QINU`"'"></span></h2></div> <p>Sorry if I'm just being stupid here, which is probably the case. </p><p>Shouldn't the coefficent of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"/></span> (underneath the first equation box in the definition section) be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>P</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc5410727492841b0fb59a144d17c6d7aae244b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.582ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{P}}}"/></span> rather than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/decd68a69fcb9a2a5c159842ef98df1f9c0d1b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.582ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{P}}}"/></span>. This feels like an easy mistake to make when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"/></span> is normally going to be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"/></span>. </p></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Talk:Fourier_series&oldid=960025892">https://en.wikipedia.org/w/index.php?title=Talk:Fourier_series&oldid=960025892</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:B-Class_mathematics_articles" title="Category:B-Class mathematics articles">B-Class mathematics articles</a></li><li><a href="/wiki/Category:Top-priority_mathematics_articles" title="Category:Top-priority mathematics articles">Top-priority mathematics articles</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Pages_using_WikiProject_Mathematics_with_unknown_parameters" title="Category:Pages using WikiProject Mathematics with unknown parameters">Pages using WikiProject Mathematics with unknown parameters</a></li><li><a href="/wiki/Category:Articles_with_WikiProject_banners_but_without_a_banner_shell" title="Category:Articles with WikiProject banners but without a banner shell">Articles with WikiProject banners but without a banner shell</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 31 May 2020, at 19:16<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright"><span class="nowraplinks">This version of the page has been <a href="/wiki/Help:Page_history" title="Help:Page history">revised</a>. 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