Loading... !!! <!doctype html> <html> <head> <meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'> <link href='https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text/css' /><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color:#ffffff; --text-color:#333333; --select-text-bg-color:#B5D6FC; --select-text-font-color:auto; --monospace:"Lucida Console",Consolas,"Courier",monospace; --title-bar-height:20px; } .mac-os-11 { --title-bar-height:28px; } html { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; -webkit-font-smoothing: antialiased; } body { margin: 0px; padding: 0px; height: auto; inset: 0px; font-size: 1rem; line-height: 1.42857; overflow-x: hidden; background: inherit; tab-size: 4; } iframe { margin: auto; } a.url { word-break: break-all; } a:active, a:hover { outline: 0px; } .in-text-selection, ::selection { text-shadow: none; background: var(--select-text-bg-color); color: var(--select-text-font-color); } #write { margin: 0px auto; height: auto; width: inherit; word-break: normal; overflow-wrap: break-word; position: relative; white-space: normal; overflow-x: visible; padding-top: 36px; } #write.first-line-indent p { text-indent: 2em; } #write.first-line-indent li p, #write.first-line-indent p * { text-indent: 0px; } #write.first-line-indent li { margin-left: 2em; } .for-image #write { padding-left: 8px; padding-right: 8px; } body.typora-export { padding-left: 30px; padding-right: 30px; } .typora-export .footnote-line, .typora-export li, .typora-export p { white-space: pre-wrap; } .typora-export .task-list-item input { pointer-events: none; } @media screen and (max-width: 500px) { body.typora-export { padding-left: 0px; padding-right: 0px; } #write { padding-left: 20px; padding-right: 20px; } .CodeMirror-sizer { margin-left: 0px !important; } .CodeMirror-gutters { display: none !important; } } #write li > figure:last-child { margin-bottom: 0.5rem; } #write ol, #write ul { position: relative; } img { max-width: 100%; vertical-align: middle; image-orientation: from-image; } button, input, select, textarea { color: inherit; font: inherit; } input[type="checkbox"], input[type="radio"] { line-height: normal; padding: 0px; } *, ::after, ::before { box-sizing: border-box; } #write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p, #write pre { width: inherit; } #write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p { position: relative; } p { line-height: inherit; } h1, h2, h3, h4, h5, h6 { break-after: avoid-page; break-inside: avoid; orphans: 4; } p { orphans: 4; } h1 { font-size: 2rem; } h2 { font-size: 1.8rem; } h3 { font-size: 1.6rem; } h4 { font-size: 1.4rem; } h5 { font-size: 1.2rem; } h6 { font-size: 1rem; } .md-math-block, .md-rawblock, h1, h2, h3, h4, h5, h6, p { margin-top: 1rem; margin-bottom: 1rem; } .hidden { display: none; } .md-blockmeta { color: rgb(204, 204, 204); font-weight: 700; font-style: italic; } a { cursor: pointer; } sup.md-footnote { padding: 2px 4px; background-color: rgba(238, 238, 238, 0.7); color: rgb(85, 85, 85); border-radius: 4px; cursor: pointer; } sup.md-footnote a, sup.md-footnote a:hover { color: inherit; text-transform: inherit; text-decoration: inherit; } #write input[type="checkbox"] { cursor: pointer; width: inherit; height: inherit; } figure { overflow-x: auto; margin: 1.2em 0px; max-width: calc(100% + 16px); padding: 0px; } figure > table { margin: 0px; } tr { break-inside: avoid; break-after: auto; } thead { display: table-header-group; } table { border-collapse: collapse; border-spacing: 0px; width: 100%; overflow: auto; break-inside: auto; text-align: left; } table.md-table td { min-width: 32px; } .CodeMirror-gutters { border-right: 0px; background-color: inherit; } .CodeMirror-linenumber { user-select: none; } .CodeMirror { text-align: left; } .CodeMirror-placeholder { opacity: 0.3; } .CodeMirror pre { padding: 0px 4px; } .CodeMirror-lines { padding: 0px; } div.hr:focus { cursor: none; } #write pre { white-space: pre-wrap; } #write.fences-no-line-wrapping pre { white-space: pre; } #write pre.ty-contain-cm { white-space: normal; } .CodeMirror-gutters { margin-right: 4px; } .md-fences { font-size: 0.9rem; display: block; break-inside: avoid; text-align: left; overflow: visible; white-space: pre; background: inherit; position: relative !important; } .md-fences-adv-panel { width: 100%; margin-top: 10px; text-align: center; padding-top: 0px; padding-bottom: 8px; overflow-x: auto; } #write .md-fences.mock-cm { white-space: pre-wrap; } .md-fences.md-fences-with-lineno { padding-left: 0px; } #write.fences-no-line-wrapping .md-fences.mock-cm { white-space: pre; overflow-x: auto; } .md-fences.mock-cm.md-fences-with-lineno { padding-left: 8px; } .CodeMirror-line, twitterwidget { break-inside: avoid; } .footnotes { opacity: 0.8; font-size: 0.9rem; margin-top: 1em; margin-bottom: 1em; } .footnotes + .footnotes { margin-top: 0px; } .md-reset { margin: 0px; padding: 0px; border: 0px; outline: 0px; vertical-align: top; background: 0px 0px; text-decoration: none; text-shadow: none; float: none; position: static; width: auto; height: auto; white-space: nowrap; cursor: inherit; -webkit-tap-highlight-color: transparent; line-height: normal; font-weight: 400; text-align: left; box-sizing: content-box; direction: ltr; } li div { padding-top: 0px; } blockquote { margin: 1rem 0px; } li .mathjax-block, li p { margin: 0.5rem 0px; } li blockquote { margin: 1rem 0px; } li { margin: 0px; position: relative; } blockquote > :last-child { margin-bottom: 0px; } blockquote > :first-child, li > :first-child { margin-top: 0px; } .footnotes-area { color: rgb(136, 136, 136); margin-top: 0.714rem; padding-bottom: 0.143rem; white-space: normal; } #write .footnote-line { white-space: pre-wrap; } @media print { body, html { border: 1px solid transparent; height: 99%; break-after: avoid; break-before: avoid; font-variant-ligatures: no-common-ligatures; } #write { margin-top: 0px; padding-top: 0px; border-color: transparent !important; } .typora-export * { -webkit-print-color-adjust: exact; } .typora-export #write { break-after: avoid; } .typora-export #write::after { height: 0px; } .is-mac table { break-inside: avoid; } .typora-export-show-outline .typora-export-sidebar { display: none; } } .footnote-line { margin-top: 0.714em; font-size: 0.7em; } a img, img a { cursor: pointer; } pre.md-meta-block { font-size: 0.8rem; min-height: 0.8rem; white-space: pre-wrap; background: rgb(204, 204, 204); display: block; overflow-x: hidden; } p > .md-image:only-child:not(.md-img-error) img, p > img:only-child { display: block; margin: auto; } #write.first-line-indent p > .md-image:only-child:not(.md-img-error) img { left: -2em; position: relative; } p > .md-image:only-child { display: inline-block; width: 100%; } #write .MathJax_Display { margin: 0.8em 0px 0px; } .md-math-block { width: 100%; } .md-math-block:not(:empty)::after { display: none; } .MathJax_ref { fill: currentcolor; } [contenteditable="true"]:active, [contenteditable="true"]:focus, [contenteditable="false"]:active, [contenteditable="false"]:focus { outline: 0px; box-shadow: none; } .md-task-list-item { position: relative; list-style-type: none; } .task-list-item.md-task-list-item { padding-left: 0px; } .md-task-list-item > input { position: absolute; top: 0px; left: 0px; margin-left: -1.2em; margin-top: calc(1em - 10px); border: none; } .math { font-size: 1rem; } .md-toc { min-height: 3.58rem; position: relative; font-size: 0.9rem; border-radius: 10px; } .md-toc-content { position: relative; margin-left: 0px; } .md-toc-content::after, .md-toc::after { display: none; } .md-toc-item { display: block; color: rgb(65, 131, 196); } .md-toc-item a { text-decoration: none; } .md-toc-inner:hover { text-decoration: underline; } .md-toc-inner { display: inline-block; cursor: pointer; } .md-toc-h1 .md-toc-inner { margin-left: 0px; font-weight: 700; } .md-toc-h2 .md-toc-inner { margin-left: 2em; } .md-toc-h3 .md-toc-inner { margin-left: 4em; } .md-toc-h4 .md-toc-inner { margin-left: 6em; } .md-toc-h5 .md-toc-inner { margin-left: 8em; } .md-toc-h6 .md-toc-inner { margin-left: 10em; } @media screen and (max-width: 48em) { .md-toc-h3 .md-toc-inner { margin-left: 3.5em; } .md-toc-h4 .md-toc-inner { margin-left: 5em; } .md-toc-h5 .md-toc-inner { margin-left: 6.5em; } .md-toc-h6 .md-toc-inner { margin-left: 8em; } } a.md-toc-inner { font-size: inherit; font-style: inherit; font-weight: inherit; line-height: inherit; } .footnote-line a:not(.reversefootnote) { color: inherit; } .md-attr { display: none; } .md-fn-count::after { content: "."; } code, pre, samp, tt { font-family: var(--monospace); } kbd { margin: 0px 0.1em; padding: 0.1em 0.6em; font-size: 0.8em; color: rgb(36, 39, 41); background: rgb(255, 255, 255); border: 1px solid rgb(173, 179, 185); border-radius: 3px; box-shadow: rgba(12, 13, 14, 0.2) 0px 1px 0px, rgb(255, 255, 255) 0px 0px 0px 2px inset; white-space: nowrap; vertical-align: middle; } .md-comment { color: rgb(162, 127, 3); opacity: 0.6; font-family: var(--monospace); } code { text-align: left; vertical-align: initial; } a.md-print-anchor { white-space: pre !important; border-width: initial !important; border-style: none !important; border-color: initial !important; display: inline-block !important; position: absolute !important; width: 1px !important; right: 0px !important; outline: 0px !important; background: 0px 0px !important; text-decoration: initial !important; text-shadow: initial !important; } .os-windows.monocolor-emoji .md-emoji { font-family: "Segoe UI Symbol", sans-serif; } .md-diagram-panel > svg { max-width: 100%; } [lang="flow"] svg, [lang="mermaid"] svg { max-width: 100%; height: auto; } [lang="mermaid"] .node text { font-size: 1rem; } table tr th { border-bottom: 0px; } video { max-width: 100%; display: block; margin: 0px auto; } iframe { max-width: 100%; width: 100%; border: none; } .highlight td, .highlight tr { border: 0px; } mark { background: rgb(255, 255, 0); color: rgb(0, 0, 0); } .md-html-inline .md-plain, .md-html-inline strong, mark .md-inline-math, mark strong { color: inherit; } .md-expand mark .md-meta { opacity: 0.3 !important; } mark .md-meta { color: rgb(0, 0, 0); } @media print { .typora-export h1, .typora-export h2, .typora-export h3, .typora-export h4, .typora-export h5, .typora-export h6 { break-inside: avoid; } } .md-diagram-panel .messageText { stroke: none !important; } .md-diagram-panel .start-state { fill: var(--node-fill); } .md-diagram-panel .edgeLabel rect { opacity: 1 !important; } .md-fences.md-fences-math { font-size: 1em; } .md-fences-advanced:not(.md-focus) { padding: 0px; white-space: nowrap; border: 0px; } .md-fences-advanced:not(.md-focus) { background: inherit; } .typora-export-show-outline .typora-export-content { max-width: 1440px; margin: auto; display: flex; flex-direction: row; } .typora-export-sidebar { width: 300px; font-size: 0.8rem; margin-top: 80px; margin-right: 18px; } .typora-export-show-outline #write { --webkit-flex:2; flex: 2 1 0%; } .typora-export-sidebar .outline-content { position: fixed; top: 0px; max-height: 100%; overflow: hidden auto; padding-bottom: 30px; padding-top: 60px; width: 300px; } @media screen and (max-width: 1024px) { .typora-export-sidebar, .typora-export-sidebar .outline-content { width: 240px; } } @media screen and (max-width: 800px) { .typora-export-sidebar { display: none; } } .outline-content li, .outline-content ul { margin-left: 0px; margin-right: 0px; padding-left: 0px; padding-right: 0px; list-style: none; } .outline-content ul { margin-top: 0px; margin-bottom: 0px; } .outline-content strong { font-weight: 400; } .outline-expander { width: 1rem; height: 1.42857rem; position: relative; display: table-cell; vertical-align: middle; cursor: pointer; padding-left: 4px; } .outline-expander::before { content: ""; position: relative; font-family: Ionicons; display: inline-block; font-size: 8px; vertical-align: middle; } .outline-item { padding-top: 3px; padding-bottom: 3px; cursor: pointer; } .outline-expander:hover::before { content: ""; } .outline-h1 > .outline-item { padding-left: 0px; } .outline-h2 > .outline-item { padding-left: 1em; } .outline-h3 > .outline-item { padding-left: 2em; } .outline-h4 > .outline-item { padding-left: 3em; } .outline-h5 > .outline-item { padding-left: 4em; } .outline-h6 > .outline-item { padding-left: 5em; } .outline-label { cursor: pointer; display: table-cell; vertical-align: middle; text-decoration: none; color: inherit; } .outline-label:hover { text-decoration: underline; } .outline-item:hover { border-color: rgb(245, 245, 245); background-color: var(--item-hover-bg-color); } .outline-item:hover { margin-left: -28px; margin-right: -28px; border-left: 28px solid transparent; border-right: 28px solid transparent; } .outline-item-single .outline-expander::before, .outline-item-single .outline-expander:hover::before { display: none; } .outline-item-open > .outline-item > .outline-expander::before { content: ""; } .outline-children { display: none; } .info-panel-tab-wrapper { display: none; } .outline-item-open > .outline-children { display: block; } .typora-export .outline-item { padding-top: 1px; padding-bottom: 1px; } .typora-export .outline-item:hover { margin-right: -8px; border-right: 8px solid transparent; } .typora-export .outline-expander::before { content: "+"; font-family: inherit; top: -1px; } .typora-export .outline-expander:hover::before, .typora-export .outline-item-open > .outline-item > .outline-expander::before { content: "−"; } .typora-export-collapse-outline .outline-children { display: none; } .typora-export-collapse-outline .outline-item-open > .outline-children, .typora-export-no-collapse-outline .outline-children { display: block; } .typora-export-no-collapse-outline .outline-expander::before { content: "" !important; } .typora-export-show-outline .outline-item-active > .outline-item .outline-label { font-weight: 700; } .md-inline-math-container mjx-container { zoom: 0.95; } :root { --side-bar-bg-color: #fafafa; --control-text-color: #777; } @include-when-export url(https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext); /* open-sans-regular - latin-ext_latin */ /* open-sans-italic - latin-ext_latin */ /* open-sans-700 - latin-ext_latin */ /* open-sans-700italic - latin-ext_latin */ html { font-size: 16px; -webkit-font-smoothing: antialiased; } body { font-family: "Open Sans","Clear Sans", "Helvetica Neue", Helvetica, Arial, 'Segoe UI Emoji', sans-serif; color: rgb(51, 51, 51); line-height: 1.6; } #write { max-width: 860px; margin: 0 auto; padding: 30px; padding-bottom: 100px; } @media only screen and (min-width: 1400px) { #write { max-width: 1024px; } } @media only screen and (min-width: 1800px) { #write { max-width: 1200px; } } #write > ul:first-child, #write > ol:first-child{ margin-top: 30px; } a { color: #4183C4; } h1, h2, h3, h4, h5, h6 { position: relative; margin-top: 1rem; margin-bottom: 1rem; font-weight: bold; line-height: 1.4; cursor: text; } h1:hover a.anchor, h2:hover a.anchor, h3:hover a.anchor, h4:hover a.anchor, h5:hover a.anchor, h6:hover a.anchor { text-decoration: none; } h1 tt, h1 code { font-size: inherit; } h2 tt, h2 code { font-size: inherit; } h3 tt, h3 code { font-size: inherit; } h4 tt, h4 code { font-size: inherit; } h5 tt, h5 code { font-size: inherit; } h6 tt, h6 code { font-size: inherit; } h1 { font-size: 2.25em; line-height: 1.2; border-bottom: 1px solid #eee; } h2 { font-size: 1.75em; line-height: 1.225; border-bottom: 1px solid #eee; } /*@media print { .typora-export h1, .typora-export h2 { border-bottom: none; padding-bottom: initial; } .typora-export h1::after, .typora-export h2::after { content: ""; display: block; height: 100px; margin-top: -96px; border-top: 1px solid #eee; } }*/ h3 { font-size: 1.5em; line-height: 1.43; } h4 { font-size: 1.25em; } h5 { font-size: 1em; } h6 { font-size: 1em; color: #777; } p, blockquote, ul, ol, dl, table{ margin: 0.8em 0; } li>ol, li>ul { margin: 0 0; } hr { height: 2px; padding: 0; margin: 16px 0; background-color: #e7e7e7; border: 0 none; overflow: hidden; box-sizing: content-box; } li p.first { display: inline-block; } ul, ol { padding-left: 30px; } ul:first-child, ol:first-child { margin-top: 0; } ul:last-child, ol:last-child { margin-bottom: 0; } blockquote { border-left: 4px solid #dfe2e5; padding: 0 15px; color: #777777; } blockquote blockquote { padding-right: 0; } table { padding: 0; word-break: initial; } table tr { border: 1px solid #dfe2e5; margin: 0; padding: 0; } table tr:nth-child(2n), thead { background-color: #f8f8f8; } table th { font-weight: bold; border: 1px solid #dfe2e5; border-bottom: 0; margin: 0; padding: 6px 13px; } table td { border: 1px solid #dfe2e5; margin: 0; padding: 6px 13px; } table th:first-child, table td:first-child { margin-top: 0; } table th:last-child, table td:last-child { margin-bottom: 0; } .CodeMirror-lines { padding-left: 4px; } .code-tooltip { box-shadow: 0 1px 1px 0 rgba(0,28,36,.3); border-top: 1px solid #eef2f2; } .md-fences, code, tt { border: 1px solid #e7eaed; background-color: #f8f8f8; border-radius: 3px; padding: 0; padding: 2px 4px 0px 4px; font-size: 0.9em; } code { background-color: #f3f4f4; padding: 0 2px 0 2px; } .md-fences { margin-bottom: 15px; margin-top: 15px; padding-top: 8px; padding-bottom: 6px; } .md-task-list-item > input { margin-left: -1.3em; } @media print { html { font-size: 13px; } table, pre { page-break-inside: avoid; } pre { word-wrap: break-word; } } .md-fences { background-color: #f8f8f8; } #write pre.md-meta-block { padding: 1rem; font-size: 85%; line-height: 1.45; background-color: #f7f7f7; border: 0; border-radius: 3px; color: #777777; margin-top: 0 !important; } .mathjax-block>.code-tooltip { bottom: .375rem; } .md-mathjax-midline { background: #fafafa; } #write>h3.md-focus:before{ left: -1.5625rem; top: .375rem; } #write>h4.md-focus:before{ left: -1.5625rem; top: .285714286rem; } #write>h5.md-focus:before{ left: -1.5625rem; top: .285714286rem; } #write>h6.md-focus:before{ left: -1.5625rem; top: .285714286rem; } .md-image>.md-meta { /*border: 1px solid #ddd;*/ border-radius: 3px; padding: 2px 0px 0px 4px; font-size: 0.9em; color: inherit; } .md-tag { color: #a7a7a7; opacity: 1; } .md-toc { margin-top:20px; padding-bottom:20px; } .sidebar-tabs { border-bottom: none; } #typora-quick-open { border: 1px solid #ddd; background-color: #f8f8f8; } #typora-quick-open-item { background-color: #FAFAFA; border-color: #FEFEFE #e5e5e5 #e5e5e5 #eee; border-style: solid; border-width: 1px; } /** focus mode */ .on-focus-mode blockquote { border-left-color: rgba(85, 85, 85, 0.12); } header, .context-menu, .megamenu-content, footer{ font-family: "Segoe UI", "Arial", sans-serif; } .file-node-content:hover .file-node-icon, .file-node-content:hover .file-node-open-state{ visibility: visible; } .mac-seamless-mode #typora-sidebar { background-color: #fafafa; background-color: var(--side-bar-bg-color); } .md-lang { color: #b4654d; } /*.html-for-mac { --item-hover-bg-color: #E6F0FE; }*/ #md-notification .btn { border: 0; } .dropdown-menu .divider { border-color: #e5e5e5; opacity: 0.4; } .ty-preferences .window-content { background-color: #fafafa; } .ty-preferences .nav-group-item.active { color: white; background: #999; } .menu-item-container a.menu-style-btn { background-color: #f5f8fa; background-image: linear-gradient( 180deg , hsla(0, 0%, 100%, 0.8), hsla(0, 0%, 100%, 0)); } mjx-container[jax="SVG"] { direction: ltr; } mjx-container[jax="SVG"] > svg { overflow: visible; min-height: 1px; min-width: 1px; } mjx-container[jax="SVG"] > svg a { fill: blue; stroke: blue; } mjx-assistive-mml { position: absolute !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); padding: 1px 0px 0px 0px !important; border: 0px !important; display: block !important; width: auto !important; overflow: hidden !important; -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none; } mjx-assistive-mml[display="block"] { width: 100% !important; } mjx-container[jax="SVG"][display="true"] { display: block; text-align: center; margin: 1em 0; } mjx-container[jax="SVG"][display="true"][width="full"] { display: flex; } mjx-container[jax="SVG"][justify="left"] { text-align: left; } mjx-container[jax="SVG"][justify="right"] { text-align: right; } g[data-mml-node="merror"] > g { fill: red; stroke: red; } g[data-mml-node="merror"] > rect[data-background] { fill: yellow; stroke: none; } g[data-mml-node="mtable"] > line[data-line], svg[data-table] > g > line[data-line] { stroke-width: 70px; fill: none; } g[data-mml-node="mtable"] > rect[data-frame], svg[data-table] > g > rect[data-frame] { stroke-width: 70px; fill: none; } g[data-mml-node="mtable"] > .mjx-dashed, svg[data-table] > g > .mjx-dashed { stroke-dasharray: 140; } g[data-mml-node="mtable"] > .mjx-dotted, svg[data-table] > g > .mjx-dotted { stroke-linecap: round; stroke-dasharray: 0,140; } g[data-mml-node="mtable"] > g > svg { overflow: visible; } [jax="SVG"] mjx-tool { display: inline-block; position: relative; width: 0; height: 0; } [jax="SVG"] mjx-tool > mjx-tip { position: absolute; top: 0; left: 0; } mjx-tool > mjx-tip { display: inline-block; padding: .2em; border: 1px solid #888; font-size: 70%; background-color: #F8F8F8; color: black; box-shadow: 2px 2px 5px #AAAAAA; } g[data-mml-node="maction"][data-toggle] { cursor: pointer; } mjx-status { display: block; position: fixed; left: 1em; bottom: 1em; min-width: 25%; padding: .2em .4em; border: 1px solid #888; font-size: 90%; background-color: #F8F8F8; color: black; } foreignObject[data-mjx-xml] { font-family: initial; line-height: normal; overflow: visible; } mjx-container[jax="SVG"] path[data-c], mjx-container[jax="SVG"] use[data-c] { stroke-width: 3; } g[data-mml-node="xypic"] path { stroke-width: inherit; } .MathJax g[data-mml-node="xypic"] path { stroke-width: inherit; } mjx-container[jax="SVG"] path[data-c], mjx-container[jax="SVG"] use[data-c] { stroke-width: 0; } </style><title>傅里叶分析</title> </head> <body class='typora-export os-windows'><div class='typora-export-content'> <div id='write' class=''><h1 id='傅里叶分析'><span>傅里叶分析</span></h1><h2 id='傅里叶级数'><span>傅里叶级数</span></h2><h4 id='1周期函数的基本性质'><span>1.周期函数的基本性质</span></h4><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n4" cid="n4" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: 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stretchy="false">)</mo><mi>x</mi><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo stretchy="false">)</mo><mi>x</mi><mo stretchy="false">]</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>d</mi><mi>x</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><h3 id='周期函数的fourier级数'><span>周期函数的Fourier级数</span></h3><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n12" cid="n12" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="69.396ex" height="73.148ex" role="img" focusable="false" viewBox="0 -16415.7 30673.2 32331.3" 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stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext>可</mtext><mtext>以</mtext><mtext>展</mtext><mtext>开</mtext><mtext>为</mtext></mtd></mtr><mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><msub><mi>a</mi><mn>0</mn></msub><mn>2</mn></mfrac><mo>+</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mi>c</mi><mi>o</mi><mi>s</mi><mi>n</mi><mi>x</mi><mo>+</mo><msub><mi>b</mi><mi>n</mi></msub><mi>s</mi><mi>i</mi><mi>n</mi><mi>n</mi><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext>则</mtext><mtext>可</mtext><mtext>以</mtext><mtext>推</mtext><mtext>出</mtext></mtd></mtr><mtr><mtd><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>π</mi></mfrac><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>π</mi></mrow><mi>π</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>n</mi><mi>x</mi><mi>d</mi><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>⋯</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><msub><mi>b</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>π</mi></mfrac><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>π</mi></mrow><mi>π</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>n</mi><mi>x</mi><mi>d</mi><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>⋯</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext>证</mtext><mtext>明</mtext><mo>:</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>π</mi></mrow><mi>π</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>m</mi><mi>x</mi><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>π</mi></mrow><mi>π</mi></msubsup><mo stretchy="false">[</mo><mfrac><msub><mi>a</mi><mn>0</mn></msub><mn>2</mn></mfrac><mo>+</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mi>c</mi><mi>o</mi><mi>s</mi><mi>n</mi><mi>x</mi><mo>+</mo><msub><mi>b</mi><mi>n</mi></msub><mi>s</mi><mi>i</mi><mi>n</mi><mi>n</mi><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>m</mi><mi>x</mi><mi>d</mi><mi>x</mi></mtd></mtr><mtr><mtd><mo>=</mo><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>π</mi></mrow><mi>π</mi></msubsup><mfrac><msub><mi>a</mi><mn>0</mn></msub><mn>2</mn></mfrac><mi>c</mi><mi>o</mi><mi>s</mi><mi>m</mi><mi>x</mi><mi>d</mi><mi>x</mi><mo>+</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>π</mi></mrow><mi>π</mi></msubsup><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mi>c</mi><mi>o</mi><mi>s</mi><mi>n</mi><mi>x</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>m</mi><mi>x</mi><mo>+</mo><msub><mi>b</mi><mi>n</mi></msub><mi>s</mi><mi>i</mi><mi>n</mi><mi>n</mi><mi>x</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>m</mi><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mtd></mtr><mtr><mtd><mi>s</mi><mi>i</mi><mi>n</mi><mi>n</mi><mi>x</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>m</mi><mi>x</mi><mtext>是</mtext><mtext>奇</mtext><mtext>函</mtext><mtext>数</mtext><mo>,</mo><mtext>积</mtext><mtext>分</mtext><mtext>为</mtext><mn>0</mn><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>m</mi><mi>x</mi><mtext>在</mtext><mtext>一</mtext><mtext>个</mtext><mtext>周</mtext><mtext>期</mtext><mtext>内</mtext><mtext>积</mtext><mtext>分</mtext><mtext>也</mtext><mtext>是</mtext><mn>0</mn></mtd></mtr><mtr><mtd><mo stretchy="false">⇒=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover><msub><mi>a</mi><mi>n</mi></msub><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>π</mi></mrow><mi>π</mi></msubsup><mi>c</mi><mi>o</mi><mi>s</mi><mi>n</mi><mi>x</mi><mi>c</mi><mi>o</mi><mi>s</mi><mi>m</mi><mi>x</mi><mi>d</mi><mi>x</mi></mtd></mtr><mtr><mtd><mtext>只</mtext><mtext>有</mtext><mtext>当</mtext><mi>n</mi><mo>=</mo><mi>m</mi><mtext>时</mtext><mo>,</mo><mtext>积</mtext><mtext>分</mtext><mtext>为</mtext><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mtext>否</mtext><mtext>则</mtext><mtext>为</mtext><mn>0</mn></mtd></mtr><mtr><mtd><mo stretchy="false">⇒=</mo><mn>2</mn><mi>π</mi><mo>⋅</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>a</mi><mi>n</mi></msub><mo>=</mo><mi>π</mi><msub><mi>a</mi><mi>n</mi></msub></mtd></mtr><mtr><mtd><mtext>同</mtext><mtext>理</mtext><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>π</mi></mrow><mi>π</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>s</mi><mi>i</mi><mi>n</mi><mi>m</mi><mi>x</mi><mi>d</mi><mi>x</mi><mo>=</mo><mi>π</mi><msub><mi>b</mi><mi>n</mi></msub></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p><span>条件:f(x)可积且绝对可积(对于这本书,只用判断可积就可以了)</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n14" cid="n14" mdtype="math_block" data-math-tag-before="0" data-math-tag-after="0" data-math-labels="[]"><div class="md-rawblock-container md-math-container" tabindex="-1"><mjx-container class="MathJax" jax="SVG" display="true" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="23.561ex" height="15.609ex" role="img" focusable="false" viewBox="0 -3699.5 10413.8 6899" xmlns:xlink="http://www.w3.org/1999/xlink" 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data-mml-node="mtext" transform="translate(1900,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="825.5px" font-family="serif">不</text></g><g data-mml-node="mtext" transform="translate(2725.6,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="825.5px" font-family="serif">可</text></g><g data-mml-node="mtext" transform="translate(3551.1,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="825.5px" font-family="serif">积</text></g><g data-mml-node="mo" transform="translate(4376.7,0)"><use data-c="2C" xlink:href="#MJX-7-TEX-N-2C"></use></g><g data-mml-node="mtext" transform="translate(4821.4,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="825.5px" font-family="serif">但</text></g><g data-mml-node="mtext" transform="translate(5646.9,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="825.5px" font-family="serif">是</text></g><g data-mml-node="mtext" transform="translate(6472.5,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="825.5px" font-family="serif">绝</text></g><g data-mml-node="mtext" transform="translate(7298.1,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="825.5px" font-family="serif">对</text></g><g data-mml-node="mtext" transform="translate(8123.6,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="825.5px" font-family="serif">可</text></g><g data-mml-node="mtext" transform="translate(8949.2,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="825.5px" font-family="serif">积</text></g></g></g><g data-mml-node="mtr" transform="translate(0,-1449.5)"><g data-mml-node="mtd" transform="translate(301.3,0)"><g data-mml-node="mtext"><text data-variant="normal" transform="scale(1,-1)" font-size="825.5px" font-family="serif">黎</text></g><g data-mml-node="mtext" transform="translate(825.6,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="825.5px" font-family="serif">曼</text></g><g data-mml-node="mtext" 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style="vertical-align: -0.025ex;"><defs><path id="MJX-12-TEX-I-1D70B" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D70B" xlink:href="#MJX-12-TEX-I-1D70B"></use></g></g></g></svg><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></mjx-assistive-mml></mjx-container><script type="math/tex">\pi</script><span>为周期,可积且绝对可积</span> 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stretchy="false">)</mo><mo>=</mo><mfrac><msub><mi>a</mi><mn>0</mn></msub><mn>2</mn></mfrac><mo>+</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover><msub><mi>a</mi><mi>n</mi></msub><mfrac><mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>n</mi><mi>w</mi><mi>x</mi></mrow></msup><mo>−</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>i</mi><mi>n</mi><mi>w</mi><mi>x</mi></mrow></msup></mrow><mn>2</mn></mfrac><mo>+</mo><msub><mi>b</mi><mi>n</mi></msub><mfrac><mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>n</mi><mi>w</mi><mi>x</mi></mrow></msup><mo>+</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>i</mi><mi>n</mi><mi>w</mi><mi>x</mi></mrow></msup></mrow><mrow><mn>2</mn><mi>i</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><msub><mi>a</mi><mn>0</mn></msub><mn>2</mn></mfrac><mo>+</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>−</mo><mi>i</mi><msub><mi>b</mi><mi>n</mi></msub></mrow><mn>2</mn></mfrac><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>n</mi><mi>w</mi><mi>x</mi></mrow></msup><mo>+</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mfrac><mrow><msub><mi>a</mi><mi>n</mi></msub><mo>+</mo><mi>i</mi><msub><mi>b</mi><mi>n</mi></msub></mrow><mn>2</mn></mfrac><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>i</mi><mi>n</mi><mi>w</mi><mi>x</mi></mrow></msup></mtd></mtr><mtr><mtd><mtext>或</mtext><mtext>者</mtext><mtext>写</mtext><mtext>成</mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></munderover><msub><mi>F</mi><mi>n</mi></msub><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>n</mi><mi>w</mi><mi>x</mi></mrow></msup></mtd></mtr><mtr><mtd><mtext>其</mtext><mtext>中</mtext><msub><mi>F</mi><mn>0</mn></msub><mo>=</mo><mfrac><msub><mi>a</mi><mn>0</mn></msub><mn>2</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>e</mi></mrow></mfrac><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>l</mi></mrow><mi>l</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mtd></mtr><mtr><mtd><msub><mi>F</mi><mi>n</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mo>±</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>e</mi></mrow></mfrac><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>e</mi></mrow><mi>e</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>n</mi><mi>w</mi><mi>x</mi><mo>±</mo><mi>i</mi><mi>s</mi><mi>i</mi><mi>n</mi><mi>n</mi><mi>w</mi><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>e</mi></mrow></mfrac><msubsup><mo data-mjx-texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>e</mi></mrow><mi>e</mi></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>i</mi><mi>n</mi><mi>w</mi><mi>x</mi></mrow></msup><mi>d</mi><mi>x</mi></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></div></div><p> </p><h3 id='例1213'><span>例12.1.3</span></h3><p><img src="https://heaticy-1310163554.cos.ap-shanghai.myqcloud.com/markdown/image-20220512103327606.png" alt="image-20220512103327606" style="zoom:50%;" style=""></p><p><img src="https://heaticy-1310163554.cos.ap-shanghai.myqcloud.com/markdown/image-20220512103435509.png" alt="image-20220512103435509" style="zoom:50%;" style=""></p><p><span>解:</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock md-focus" id="mathjax-n26" cid="n26" mdtype="math_block" data-math-tag-before="0" data-math-labels="[]" data-math-tag-after="0"><div class="md-rawblock-container md-math-container" contenteditable="false" 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