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href="/atom.xml" title="Magicland" type="application/atom+xml"><link rel="alternate" href="/rss2.xml" title="Magicland" type="application/rss+xml"><title>离散世界与连续世界的联系 - Magicland</title><meta name="keywords" content="数学, Math,MHuiG, @MHuiG, Blog, 博客, Magicland, 魔法世界"><meta desc name="description" content="本文探讨离散与连续数学世界的联系，聚焦伽玛函数如何实现阶乘的连续化表达。通过推导伽玛函数积分形式及递推关系Γ(α+1)=αΓ(α)，证明其与阶乘的等价性Γ(n+1)=n!，并给出Γ(1/2)=√π等特殊值计算。 - MHuiG - Magicland"><meta property="og:type" content="article"><meta property="og:title" content="离散世界与连续世界的联系"><meta property="og:url" content="https://blog.mhuig.top/p/2b995a0c/"><meta property="og:site_name" content="Magicland"><meta property="og:description" content="本文探讨离散与连续数学世界的联系，聚焦伽玛函数如何实现阶乘的连续化表达。通过推导伽玛函数积分形式及递推关系Γ(α+1)=αΓ(α)，证明其与阶乘的等价性Γ(n+1)=n!，并给出Γ(1/2)=√π等特殊值计算。"><meta property="og:locale"><meta property="og:image" content="https://blog.mhuig.top/lib/favicon/android-chrome-192x192.png"><meta property="article:published_time" content="2021-01-02T12:34:00.000Z"><meta 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content="离散世界与连续世界的联系"><meta itemprop="description" content="本文探讨离散与连续数学世界的联系，聚焦伽玛函数如何实现阶乘的连续化表达。通过推导伽玛函数积分形式及递推关系Γ(α+1)=αΓ(α)，证明其与阶乘的等价性Γ(n+1)=n!，并给出Γ(1/2)=√π等特殊值计算。"></span><div class="article-meta" id="top"><h1 class="title allow-select" itemprop="name headline"> 离散世界与连续世界的联系</h1><div class="new-meta-box"><div class="new-meta-item author" itemprop="author" itemscope itemtype="http://schema.org/Person"> <a itemprop="url" class="author" target="_blank" href="https://mhuig.top/" rel="nofollow noopener" alt="author"><img itemprop="image" src="/lib/avatar/avatar-16.webp" class="lazyload" data-srcset="/lib/avatar/avatar-16.webp" srcset="data:image/gif;base64,R0lGODlhAQABAIAAAP///////yH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" alt="author"><p itemprop="name">MHuiG</p></a></div><div class="new-meta-item wordcount"><a class="notlink"><i class="fa-duotone fa-keyboard fa-fw" aria-hidden="true"></i><p>字数：183 字</p></a></div><div class="new-meta-item readtime"><a class="notlink"><i class="fa-duotone fa-hourglass-half fa-fw" aria-hidden="true"></i><p>时长：1 分钟</p></a></div></div></div><div id="layoutHelper-page-plugins"></div><div id="post-body" itemprop="articleBody"><div class="story post-story"><h2 id="探索"><a href="#探索" class="headerlink" title="探索"></a>探索</h2><p>当年，哥德巴赫异想天开就想离散的世界的阶乘<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.986ex" height="1.645ex" role="img" focusable="false" viewbox="0 -716 878 727"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="n!"><g data-mml-node="mi" data-latex="n"><path data-c="1D45B" d="M537 137C514 58 481 18 440 18C427 18 420 28 420 47C420 61 426 84 438 115C478 224 498 296 498 333C498 403 451 442 381 442C322 442 271 416 230 363C222 407 187 442 136 442C80 442 58 390 44 345C34 313 29 294 29 287C29 278 34 273 45 273C50 273 53 274 56 276C61 285 64 292 65 299C83 375 106 413 133 413C151 413 160 399 160 371C160 358 155 331 144 290L87 63C84 50 78 24 78 19C78-1 89-11 111-11C130-11 144-1 151 19C153 24 159 49 170 92L191 181L221 295C232 318 249 341 270 365C299 397 335 413 378 413C411 413 427 391 427 348C427 310 406 234 363 120C356 102 353 87 353 74C353 25 390-11 438-11C482-11 517 14 542 64C561 104 571 131 571 144C571 153 566 158 555 158C552 158 537 149 537 137Z"/></g><g data-mml-node="mo" data-latex="!" transform="translate(600,0)"><path data-c="21" d="M139 716C110 716 83 691 86 661L122 211C123 194 129 185 139 185C149 185 155 193 156 210L192 661C195 691 168 716 139 716M192 56C192 87 169 113 139 113C109 113 86 87 86 56C86 26 109 0 139 0C169 0 192 26 192 56Z"/></g></g></g></svg></mjx-container>能不能用连续世界的积分 <mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.691ex;" xmlns="http://www.w3.org/2000/svg" width="1.505ex" height="2.514ex" role="img" focusable="false" viewbox="0 -805.5 665 1111"><g stroke="currentColor" fill="currentColor" 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stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -1361)"><g data-mml-node="math" data-latex="
n!=\int?
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\Gamma(\alpha)=\int_0^{+\infty}x^{\alpha-1}e^{-x}dx \ \ \ \ \ \ \ \ \ \ (\alpha>0)
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transform="translate(2709.8,0)"><path data-c="1D6FC" d="M310 442C241 442 179 412 124 353C69 294 41 229 41 159C41 61 107-11 205-11C274-11 342 16 410 69C425 16 456-11 502-11C538-11 589 27 589 63C589 72 584 76 573 76C566 76 560 72 557 64C549 43 528 18 505 18C488 18 479 50 479 115C479 129 482 140 488 147C515 180 539 218 558 260C583 314 598 354 602 380L602 384C599 391 593 394 586 394C581 393 575 385 568 371C547 299 518 237 479 186L479 236C479 352 421 442 310 442M403 211C403 152 404 116 405 103C340 46 274 18 207 18C150 18 122 53 122 122C122 180 155 288 178 324C216 383 260 413 309 413C340 413 361 401 374 377C381 365 386 353 391 342C399 319 403 258 403 211Z"/></g><g data-mml-node="mo" data-latex=")" transform="translate(3349.8,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg></mjx-container></p><p>得到：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 27.009ex;"><svg style="vertical-align: -0.566ex; min-width: 27.009ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false" data-mjx-viewbox="0 -750 11938 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math" data-latex="
\Gamma(\alpha+1)=\alpha\Gamma(\alpha)
"><g data-mml-node="mtable" data-latex="\Gamma(\alpha+1)=\alpha\Gamma(\alpha)" transform="translate(2078,0) translate(-2078,0)"><g transform="translate(0 750) matrix(1 0 0 -1 0 0) scale(55.25)"><svg data-table="true" preserveaspectratio="xMidYMid" viewbox="3891 -750 1 1000"><g transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mlabeledtr"><g data-mml-node="mtd"><g data-mml-node="mi" data-latex="\Gamma"><path data-c="393" d="M274 641L376 641C443 641 487 626 509 596C538 555 541 520 550 450L583 450L554 680L33 680L33 641L61 641C96 641 117 639 124 634C131 629 135 617 135 599L135 81C135 63 131 51 124 46C117 41 96 39 61 39L33 39L33 0C59 2 110 3 187 3C274 3 330 2 356 0L356 39L320 39C267 39 238 44 233 55C231 60 230 69 230 82L230 606C230 641 236 641 274 641Z"/></g><g data-mml-node="mo" data-latex="(" transform="translate(625,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mi" data-latex="\alpha" transform="translate(1014,0)"><path data-c="1D6FC" d="M310 442C241 442 179 412 124 353C69 294 41 229 41 159C41 61 107-11 205-11C274-11 342 16 410 69C425 16 456-11 502-11C538-11 589 27 589 63C589 72 584 76 573 76C566 76 560 72 557 64C549 43 528 18 505 18C488 18 479 50 479 115C479 129 482 140 488 147C515 180 539 218 558 260C583 314 598 354 602 380L602 384C599 391 593 394 586 394C581 393 575 385 568 371C547 299 518 237 479 186L479 236C479 352 421 442 310 442M403 211C403 152 404 116 405 103C340 46 274 18 207 18C150 18 122 53 122 122C122 180 155 288 178 324C216 383 260 413 309 413C340 413 361 401 374 377C381 365 386 353 391 342C399 319 403 258 403 211Z"/></g><g data-mml-node="mo" data-latex="+" transform="translate(1876.2,0)"><path data-c="2B" d="M698 274L413 274L413 559C413 575 405 583 389 583C373 583 365 575 365 559L365 274L80 274C64 274 56 266 56 250C56 234 64 226 80 226L365 226L365-59C365-75 373-83 389-83C405-83 413-75 413-59L413 226L698 226C714 226 722 234 722 250C722 263 711 274 698 274Z"/></g><g data-mml-node="mn" data-latex="1" transform="translate(2876.4,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g><g data-mml-node="mo" data-latex=")" transform="translate(3376.4,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g><g data-mml-node="mo" data-latex="=" transform="translate(4043.2,0)"><path data-c="3D" d="M698 367L80 367C64 367 56 359 56 344C56 329 64 321 80 321L698 321C714 321 722 329 722 344C722 356 711 367 698 367M698 179L80 179C64 179 56 171 56 156C56 141 64 133 80 133L698 133C714 133 722 141 722 156C722 169 711 179 698 179Z"/></g><g data-mml-node="mi" data-latex="\alpha" transform="translate(5099,0)"><path data-c="1D6FC" d="M310 442C241 442 179 412 124 353C69 294 41 229 41 159C41 61 107-11 205-11C274-11 342 16 410 69C425 16 456-11 502-11C538-11 589 27 589 63C589 72 584 76 573 76C566 76 560 72 557 64C549 43 528 18 505 18C488 18 479 50 479 115C479 129 482 140 488 147C515 180 539 218 558 260C583 314 598 354 602 380L602 384C599 391 593 394 586 394C581 393 575 385 568 371C547 299 518 237 479 186L479 236C479 352 421 442 310 442M403 211C403 152 404 116 405 103C340 46 274 18 207 18C150 18 122 53 122 122C122 180 155 288 178 324C216 383 260 413 309 413C340 413 361 401 374 377C381 365 386 353 391 342C399 319 403 258 403 211Z"/></g><g data-mml-node="mi" data-latex="\Gamma" transform="translate(5739,0)"><path data-c="393" d="M274 641L376 641C443 641 487 626 509 596C538 555 541 520 550 450L583 450L554 680L33 680L33 641L61 641C96 641 117 639 124 634C131 629 135 617 135 599L135 81C135 63 131 51 124 46C117 41 96 39 61 39L33 39L33 0C59 2 110 3 187 3C274 3 330 2 356 0L356 39L320 39C267 39 238 44 233 55C231 60 230 69 230 82L230 606C230 641 236 641 274 641Z"/></g><g data-mml-node="mo" data-latex="(" transform="translate(6364,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mi" data-latex="\alpha" transform="translate(6753,0)"><path data-c="1D6FC" d="M310 442C241 442 179 412 124 353C69 294 41 229 41 159C41 61 107-11 205-11C274-11 342 16 410 69C425 16 456-11 502-11C538-11 589 27 589 63C589 72 584 76 573 76C566 76 560 72 557 64C549 43 528 18 505 18C488 18 479 50 479 115C479 129 482 140 488 147C515 180 539 218 558 260C583 314 598 354 602 380L602 384C599 391 593 394 586 394C581 393 575 385 568 371C547 299 518 237 479 186L479 236C479 352 421 442 310 442M403 211C403 152 404 116 405 103C340 46 274 18 207 18C150 18 122 53 122 122C122 180 155 288 178 324C216 383 260 413 309 413C340 413 361 401 374 377C381 365 386 353 391 342C399 319 403 258 403 211Z"/></g><g data-mml-node="mo" data-latex=")" transform="translate(7393,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></g></svg><svg data-labels="true" preserveaspectratio="xMaxYMid" viewbox="1278 -750 1 1000"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:4"><g data-mml-node="mtext" data-latex="\text{(}"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mtext" data-latex="\text{4}" transform="translate(389,0)"><path data-c="34" d="M353 677C344 677 336 672 330 663L28 199L28 163L289 163L289 81C289 63 285 51 278 46C271 41 252 39 219 39L194 39L194 0C223 2 269 3 331 3C393 3 439 2 468 0L468 39L443 39C410 39 391 41 384 46C377 51 373 63 373 81L373 163L471 163L471 202L373 202L373 660C373 670 366 677 353 677M295 553L295 202L67 202Z"/></g><g data-mml-node="mtext" data-latex="\text{)}" transform="translate(889,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>当<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="1.448ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 640 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\alpha=1"><g data-mml-node="mi" data-latex="\alpha"><path data-c="1D6FC" d="M310 442C241 442 179 412 124 353C69 294 41 229 41 159C41 61 107-11 205-11C274-11 342 16 410 69C425 16 456-11 502-11C538-11 589 27 589 63C589 72 584 76 573 76C566 76 560 72 557 64C549 43 528 18 505 18C488 18 479 50 479 115C479 129 482 140 488 147C515 180 539 218 558 260C583 314 598 354 602 380L602 384C599 391 593 394 586 394C581 393 575 385 568 371C547 299 518 237 479 186L479 236C479 352 421 442 310 442M403 211C403 152 404 116 405 103C340 46 274 18 207 18C150 18 122 53 122 122C122 180 155 288 178 324C216 383 260 413 309 413C340 413 361 401 374 377C381 365 386 353 391 342C399 319 403 258 403 211Z"/></g></g></g></svg><mjx-break size="4"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="3.52ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 1555.8 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\alpha=1"><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367L80 367C64 367 56 359 56 344C56 329 64 321 80 321L698 321C714 321 722 329 722 344C722 356 711 367 698 367M698 179L80 179C64 179 56 171 56 156C56 141 64 133 80 133L698 133C714 133 722 141 722 156C722 169 711 179 698 179Z"/></g><g data-mml-node="mn" data-latex="1" transform="translate(1055.8,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g></g></g></svg></mjx-container>时,<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.305ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 1903 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Gamma(1)=\int_0^{+\infty}e^{-x}dx=1"><g data-mml-node="mi" data-latex="\Gamma"><path data-c="393" d="M274 641L376 641C443 641 487 626 509 596C538 555 541 520 550 450L583 450L554 680L33 680L33 641L61 641C96 641 117 639 124 634C131 629 135 617 135 599L135 81C135 63 131 51 124 46C117 41 96 39 61 39L33 39L33 0C59 2 110 3 187 3C274 3 330 2 356 0L356 39L320 39C267 39 238 44 233 55C231 60 230 69 230 82L230 606C230 641 236 641 274 641Z"/></g><g data-mml-node="mo" data-latex="(" transform="translate(625,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mn" data-latex="1" transform="translate(1014,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g><g data-mml-node="mo" data-latex=")" transform="translate(1514,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg><mjx-break size="4"> </mjx-break><svg style="vertical-align: -0.806ex;" xmlns="http://www.w3.org/2000/svg" width="13.302ex" height="2.944ex" role="img" focusable="false" viewbox="0 -944.8 5879.3 1301.2"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Gamma(1)=\int_0^{+\infty}e^{-x}dx=1"><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367L80 367C64 367 56 359 56 344C56 329 64 321 80 321L698 321C714 321 722 329 722 344C722 356 711 367 698 367M698 179L80 179C64 179 56 171 56 156C56 141 64 133 80 133L698 133C714 133 722 141 722 156C722 169 711 179 698 179Z"/></g><g data-mml-node="msubsup" data-latex="\int_0^{+\infty}" transform="translate(1055.8,0)"><g data-mml-node="mo" data-latex="\int" transform="translate(0 0.5)"><path data-c="222B" d="M497 805C421 805 370 747 344 630C340 613 335 578 328 526L315 425C301 313 292 230 288 175L269-72C262-169 246-276 168-276C147-276 129-271 114-261C135-255 146-241 146-219C146-192 128-173 101-173C71-173 56-189 56-220C56-273 111-306 168-306C249-306 305-246 335-127C356-43 375 107 391 324L410 571C414 628 421 673 430 705C443 747 452 775 498 775C519 775 537 770 551 760C530 754 519 740 519 718C519 691 537 672 564 672C594 672 609 688 609 719C609 772 554 805 497 805Z"/></g><g data-mml-node="TeXAtom" transform="translate(754,532.6) scale(0.707)" data-latex="{+\infty}" data-mjx-texclass="ORD"><g data-mml-node="mo" data-latex="+"><path data-c="2B" d="M698 274L413 274L413 559C413 575 405 583 389 583C373 583 365 575 365 559L365 274L80 274C64 274 56 266 56 250C56 234 64 226 80 226L365 226L365-59C365-75 373-83 389-83C405-83 413-75 413-59L413 226L698 226C714 226 722 234 722 250C722 263 711 274 698 274Z"/></g><g data-mml-node="mi" data-latex="\infty" transform="translate(778,0)"><path data-c="221E" d="M749-11C807-11 855 13 892 60C926 104 943 156 943 216C943 275 926 327 893 371C856 418 809 442 752 442C684 442 625 416 576 364C547 332 524 303 507 278C464 329 435 361 421 373C367 419 310 442 250 442C192 442 144 418 107 371C73 327 56 275 56 215C56 156 73 104 106 60C143 13 190-11 247-11C315-11 374 15 423 67C452 99 475 128 492 153C535 102 564 70 578 58C632 12 689-11 749-11M913 216C913 188 911 168 908 156C903 137 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415 355C415 410 363 442 307 442C268 442 229 432 190 412C113 372 46 281 46 171C46 69 105-11 204-11C257-11 304 2 345 27C379 48 404 69 420 90C427 99 430 106 430 109C430 120 425 126 414 126C409 126 404 122 398 114C365 70 324 42 277 30C246 22 223 18 206 18C149 18 124 72 124 129M375 355C375 289 311 256 182 256L147 256C166 322 194 366 232 387C262 404 287 413 307 413C343 413 375 391 375 355Z"/></g><g data-mml-node="TeXAtom" transform="translate(499,363) scale(0.707)" data-latex="{-x}" data-mjx-texclass="ORD"><g data-mml-node="mo" data-latex="-"><path data-c="2212" d="M698 270L80 270C64 270 56 263 56 250C56 237 64 230 80 230L698 230C714 230 722 237 722 250C722 262 710 270 698 270Z"/></g><g data-mml-node="mi" data-latex="x" transform="translate(778,0)"><path data-c="1D465" d="M527 373C527 419 482 442 432 442C389 442 355 419 329 373C308 419 273 442 222 442C173 442 133 419 101 374C74 335 60 306 60 287C60 278 65 273 75 273C84 273 90 278 92 287C111 345 153 413 220 413C253 413 269 392 269 351C269 330 251 252 216 118C199 51 169 18 126 18C112 18 99 21 88 26C114 36 127 54 127 80C127 106 114 119 87 119C54 119 29 91 29 58C29 12 76-11 125-11C167-11 201 12 228 58C247 12 283-11 335-11C383-11 423 12 455 57C482 96 496 125 496 144C496 153 491 158 481 158C472 158 467 153 464 144C447 87 402 18 337 18C304 18 287 38 287 79C287 92 292 120 303 165L337 300C356 375 387 413 431 413C445 413 458 410 469 405C442 396 429 378 429 351C429 325 443 312 470 312C502 312 527 341 527 373Z"/></g></g></g><g data-mml-node="mi" data-latex="d" transform="translate(4787.3,0)"><path data-c="1D451" d="M429 632L370 389C349 426 320 445 281 445C216 445 159 412 109 345C63 283 40 218 40 151C40 63 91-11 175-11C218-11 260 12 300 59C311 21 345-11 392-11C461-11 483 71 498 145C498 154 493 159 483 159C474 159 468 152 465 138C445 59 421 19 394 19C377 19 368 33 368 60C368 75 370 91 374 107L516 679L516 683C513 690 507 694 499 694C476 694 434 690 374 683C363 682 357 674 357 660C357 650 366 645 384 645C403 645 429 646 429 632M341 376C351 355 356 341 356 332C355 328 354 323 353 316L304 122C301 111 294 99 285 87C248 42 212 19 177 19C138 19 118 49 118 108C118 132 124 168 136 216C157 301 187 359 224 390C244 407 263 415 282 415C309 415 329 402 341 376Z"/></g><g data-mml-node="mi" data-latex="x" transform="translate(5307.3,0)"><path data-c="1D465" d="M527 373C527 419 482 442 432 442C389 442 355 419 329 373C308 419 273 442 222 442C173 442 133 419 101 374C74 335 60 306 60 287C60 278 65 273 75 273C84 273 90 278 92 287C111 345 153 413 220 413C253 413 269 392 269 351C269 330 251 252 216 118C199 51 169 18 126 18C112 18 99 21 88 26C114 36 127 54 127 80C127 106 114 119 87 119C54 119 29 91 29 58C29 12 76-11 125-11C167-11 201 12 228 58C247 12 283-11 335-11C383-11 423 12 455 57C482 96 496 125 496 144C496 153 491 158 481 158C472 158 467 153 464 144C447 87 402 18 337 18C304 18 287 38 287 79C287 92 292 120 303 165L337 300C356 375 387 413 431 413C445 413 458 410 469 405C442 396 429 378 429 351C429 325 443 312 470 312C502 312 527 341 527 373Z"/></g></g></g></svg><mjx-break size="4"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="3.52ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 1555.8 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Gamma(1)=\int_0^{+\infty}e^{-x}dx=1"><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367L80 367C64 367 56 359 56 344C56 329 64 321 80 321L698 321C714 321 722 329 722 344C722 356 711 367 698 367M698 179L80 179C64 179 56 171 56 156C56 141 64 133 80 133L698 133C714 133 722 141 722 156C722 169 711 179 698 179Z"/></g><g data-mml-node="mn" data-latex="1" transform="translate(1055.8,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g></g></g></svg></mjx-container></p><p>当<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="1.448ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 640 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\alpha=\frac{1}{2}"><g data-mml-node="mi" data-latex="\alpha"><path data-c="1D6FC" d="M310 442C241 442 179 412 124 353C69 294 41 229 41 159C41 61 107-11 205-11C274-11 342 16 410 69C425 16 456-11 502-11C538-11 589 27 589 63C589 72 584 76 573 76C566 76 560 72 557 64C549 43 528 18 505 18C488 18 479 50 479 115C479 129 482 140 488 147C515 180 539 218 558 260C583 314 598 354 602 380L602 384C599 391 593 394 586 394C581 393 575 385 568 371C547 299 518 237 479 186L479 236C479 352 421 442 310 442M403 211C403 152 404 116 405 103C340 46 274 18 207 18C150 18 122 53 122 122C122 180 155 288 178 324C216 383 260 413 309 413C340 413 361 401 374 377C381 365 386 353 391 342C399 319 403 258 403 211Z"/></g></g></g></svg><mjx-break size="4"> </mjx-break><svg style="vertical-align: -0.781ex;" xmlns="http://www.w3.org/2000/svg" width="4.184ex" height="2.737ex" role="img" focusable="false" viewbox="0 -864.9 1849.3 1209.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\alpha=\frac{1}{2}"><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367L80 367C64 367 56 359 56 344C56 329 64 321 80 321L698 321C714 321 722 329 722 344C722 356 711 367 698 367M698 179L80 179C64 179 56 171 56 156C56 141 64 133 80 133L698 133C714 133 722 141 722 156C722 169 711 179 698 179Z"/></g><g data-mml-node="mfrac" data-latex="\frac{1}{2}" transform="translate(1055.8,0)"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)" data-latex="1"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="2"><path data-c="32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"/></g><rect width="553.6" height="60" x="120" y="220"/></g></g></g></svg></mjx-container>时,<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.781ex;" xmlns="http://www.w3.org/2000/svg" width="4.97ex" height="2.737ex" role="img" focusable="false" viewbox="0 -864.9 2196.6 1209.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Gamma(\frac{1}{2})=2\int_0^{+\infty}e^{-t^2}dt=\sqrt{\pi}"><g data-mml-node="mi" data-latex="\Gamma"><path data-c="393" d="M274 641L376 641C443 641 487 626 509 596C538 555 541 520 550 450L583 450L554 680L33 680L33 641L61 641C96 641 117 639 124 634C131 629 135 617 135 599L135 81C135 63 131 51 124 46C117 41 96 39 61 39L33 39L33 0C59 2 110 3 187 3C274 3 330 2 356 0L356 39L320 39C267 39 238 44 233 55C231 60 230 69 230 82L230 606C230 641 236 641 274 641Z"/></g><g data-mml-node="mo" data-latex="(" transform="translate(625,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mfrac" data-latex="\frac{1}{2}" transform="translate(1014,0)"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)" data-latex="1"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="2"><path data-c="32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"/></g><rect width="553.6" height="60" x="120" y="220"/></g><g data-mml-node="mo" data-latex=")" transform="translate(1807.6,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg><mjx-break size="4"> </mjx-break><svg style="vertical-align: -0.806ex;" xmlns="http://www.w3.org/2000/svg" width="14.693ex" height="2.962ex" role="img" focusable="false" viewbox="0 -952.7 6494.4 1309.1"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Gamma(\frac{1}{2})=2\int_0^{+\infty}e^{-t^2}dt=\sqrt{\pi}"><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367L80 367C64 367 56 359 56 344C56 329 64 321 80 321L698 321C714 321 722 329 722 344C722 356 711 367 698 367M698 179L80 179C64 179 56 171 56 156C56 141 64 133 80 133L698 133C714 133 722 141 722 156C722 169 711 179 698 179Z"/></g><g data-mml-node="mn" data-latex="2" transform="translate(1055.8,0)"><path data-c="32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"/></g><g data-mml-node="msubsup" data-latex="\int_0^{+\infty}" transform="translate(1722.4,0)"><g data-mml-node="mo" data-latex="\int" transform="translate(0 0.5)"><path data-c="222B" d="M497 805C421 805 370 747 344 630C340 613 335 578 328 526L315 425C301 313 292 230 288 175L269-72C262-169 246-276 168-276C147-276 129-271 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data-latex="t^2" transform="translate(778,0)"><g data-mml-node="mi" data-latex="t"><path data-c="1D461" d="M330 419C330 428 320 433 299 433L218 433C244 537 257 591 257 595C257 616 246 626 225 626C202 626 188 613 182 587L145 433L56 433C34 433 23 432 23 411C23 400 33 395 54 395L135 395C86 200 62 97 62 84C62 28 100-11 156-11C194-11 227 6 256 41C280 70 297 98 308 125C312 136 314 142 314 145C314 154 309 159 299 159C291 159 285 154 281 143C247 60 206 19 157 19C140 19 131 33 131 60C131 75 133 91 137 107L208 395L297 395C323 395 330 397 330 419Z"/></g><g data-mml-node="mn" transform="translate(394,363) scale(0.707)" data-latex="2"><path data-c="32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"/></g></g></g></g><g data-mml-node="mi" data-latex="d" transform="translate(5613.4,0)"><path data-c="1D451" d="M429 632L370 389C349 426 320 445 281 445C216 445 159 412 109 345C63 283 40 218 40 151C40 63 91-11 175-11C218-11 260 12 300 59C311 21 345-11 392-11C461-11 483 71 498 145C498 154 493 159 483 159C474 159 468 152 465 138C445 59 421 19 394 19C377 19 368 33 368 60C368 75 370 91 374 107L516 679L516 683C513 690 507 694 499 694C476 694 434 690 374 683C363 682 357 674 357 660C357 650 366 645 384 645C403 645 429 646 429 632M341 376C351 355 356 341 356 332C355 328 354 323 353 316L304 122C301 111 294 99 285 87C248 42 212 19 177 19C138 19 118 49 118 108C118 132 124 168 136 216C157 301 187 359 224 390C244 407 263 415 282 415C309 415 329 402 341 376Z"/></g><g data-mml-node="mi" data-latex="t" transform="translate(6133.4,0)"><path data-c="1D461" d="M330 419C330 428 320 433 299 433L218 433C244 537 257 591 257 595C257 616 246 626 225 626C202 626 188 613 182 587L145 433L56 433C34 433 23 432 23 411C23 400 33 395 54 395L135 395C86 200 62 97 62 84C62 28 100-11 156-11C194-11 227 6 256 41C280 70 297 98 308 125C312 136 314 142 314 145C314 154 309 159 299 159C291 159 285 154 281 143C247 60 206 19 157 19C140 19 131 33 131 60C131 75 133 91 137 107L208 395L297 395C323 395 330 397 330 419Z"/></g></g></g></svg><mjx-break size="4"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.608ex" height="2.46ex" role="img" focusable="false" viewbox="0 -837.5 2478.8 1087.5"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Gamma(\frac{1}{2})=2\int_0^{+\infty}e^{-t^2}dt=\sqrt{\pi}"><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367L80 367C64 367 56 359 56 344C56 329 64 321 80 321L698 321C714 321 722 329 722 344C722 356 711 367 698 367M698 179L80 179C64 179 56 171 56 156C56 141 64 133 80 133L698 133C714 133 722 141 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366L515 366C550 366 567 378 567 403C567 426 549 431 524 431Z"/></g></g><rect width="570" height="60" x="853" y="717.5"/></g></g></g></svg></mjx-container></p><p>如：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.305ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 1903 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Gamma(3)=\Gamma(3+1)=2\Gamma(2)=2\Gamma(1+1)=2 \cdot1\Gamma(1)=2!"><g data-mml-node="mi" data-latex="\Gamma"><path data-c="393" d="M274 641L376 641C443 641 487 626 509 596C538 555 541 520 550 450L583 450L554 680L33 680L33 641L61 641C96 641 117 639 124 634C131 629 135 617 135 599L135 81C135 63 131 51 124 46C117 41 96 39 61 39L33 39L33 0C59 2 110 3 187 3C274 3 330 2 356 0L356 39L320 39C267 39 238 44 233 55C231 60 230 69 230 82L230 606C230 641 236 641 274 641Z"/></g><g 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198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 457 117 457 173C457 267 383 332 303 353Z"/></g><g data-mml-node="mo" data-latex=")" transform="translate(1514,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg><mjx-break size="4"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.814ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 2569.8 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Gamma(3)=\Gamma(3+1)=2\Gamma(2)=2\Gamma(1+1)=2 \cdot1\Gamma(1)=2!"><g data-mml-node="mo" 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d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg><mjx-break size="4"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="6.945ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 3069.8 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Gamma(3)=\Gamma(3+1)=2\Gamma(2)=2\Gamma(1+1)=2 \cdot1\Gamma(1)=2!"><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367L80 367C64 367 56 359 56 344C56 329 64 321 80 321L698 321C714 321 722 329 722 344C722 356 711 367 698 367M698 179L80 179C64 179 56 171 56 156C56 141 64 133 80 133L698 133C714 133 722 141 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fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Gamma(3)=\Gamma(3+1)=2\Gamma(2)=2\Gamma(1+1)=2 \cdot1\Gamma(1)=2!"><g data-mml-node="mo" data-latex="+"><path data-c="2B" d="M698 274L413 274L413 559C413 575 405 583 389 583C373 583 365 575 365 559L365 274L80 274C64 274 56 266 56 250C56 234 64 226 80 226L365 226L365-59C365-75 373-83 389-83C405-83 413-75 413-59L413 226L698 226C714 226 722 234 722 250C722 263 711 274 698 274Z"/></g><g data-mml-node="mn" data-latex="1" transform="translate(1000.2,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g><g data-mml-node="mo" data-latex=")" transform="translate(1500.2,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 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transform="translate(1055.8,0)"><path data-c="32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"/></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="6.568ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 2903.2 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Gamma(3)=\Gamma(3+1)=2\Gamma(2)=2\Gamma(1+1)=2 \cdot1\Gamma(1)=2!"><g data-mml-node="mo" data-latex="\cdot"><path data-c="22C5" d="M192 250C192 279 168 303 139 303C110 303 86 279 86 250C86 221 110 197 139 197C168 197 192 221 192 250Z"/></g><g data-mml-node="mn" data-latex="1" transform="translate(500.2,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g><g data-mml-node="mi" data-latex="\Gamma" transform="translate(1000.2,0)"><path data-c="393" d="M274 641L376 641C443 641 487 626 509 596C538 555 541 520 550 450L583 450L554 680L33 680L33 641L61 641C96 641 117 639 124 634C131 629 135 617 135 599L135 81C135 63 131 51 124 46C117 41 96 39 61 39L33 39L33 0C59 2 110 3 187 3C274 3 330 2 356 0L356 39L320 39C267 39 238 44 233 55C231 60 230 69 230 82L230 606C230 641 236 641 274 641Z"/></g><g data-mml-node="mo" data-latex="(" transform="translate(1625.2,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mn" data-latex="1" transform="translate(2014.2,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47C195 42 170 39 130 39L95 39L95 0C120 2 174 3 257 3C340 3 394 2 419 0L419 39L384 39C343 39 318 42 310 47C302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"/></g><g data-mml-node="mo" data-latex=")" transform="translate(2514.2,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg><mjx-break size="4"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.149ex" height="2.262ex" role="img" focusable="false" viewbox="0 -750 1833.8 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Gamma(3)=\Gamma(3+1)=2\Gamma(2)=2\Gamma(1+1)=2 \cdot1\Gamma(1)=2!"><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367L80 367C64 367 56 359 56 344C56 329 64 321 80 321L698 321C714 321 722 329 722 344C722 356 711 367 698 367M698 179L80 179C64 179 56 171 56 156C56 141 64 133 80 133L698 133C714 133 722 141 722 156C722 169 711 179 698 179Z"/></g><g data-mml-node="mn" data-latex="2" transform="translate(1055.8,0)"><path data-c="32" d="M237 666C186 666 143 648 106 612C69 576 50 534 50 483C50 449 75 424 106 424C136 424 161 450 161 480C161 513 137 536 105 536C102 536 100 536 98 535C117 584 161 627 224 627C306 627 352 556 352 470C352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0L450 180L417 180C409 129 402 100 396 91C391 86 361 84 306 84L139 84L236 179C304 243 390 312 419 365C439 400 449 435 449 470C449 588 357 666 237 666Z"/></g><g data-mml-node="mo" data-latex="!" transform="translate(1555.8,0)"><path data-c="21" d="M139 716C110 716 83 691 86 661L122 211C123 194 129 185 139 185C149 185 155 193 156 210L192 661C195 691 168 716 139 716M192 56C192 87 169 113 139 113C109 113 86 87 86 56C86 26 109 0 139 0C169 0 192 26 192 56Z"/></g></g></g></svg></mjx-container></p><p>又如：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.781ex;" xmlns="http://www.w3.org/2000/svg" width="4.97ex" height="2.737ex" role="img" focusable="false" viewbox="0 -864.9 2196.6 1209.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Gamma(\frac{5}{2})=\Gamma(\frac{3}{2}+1)=\frac{3}{2}\Gamma(\frac{3}{2})=\frac{3}{2}\Gamma(\frac{1}{2}+1)=\frac{3}{2}\cdot\frac{1}{2}\Gamma(\frac{1}{2})=\frac{3}{4}\sqrt{\pi}"><g data-mml-node="mi" data-latex="\Gamma"><path data-c="393" d="M274 641L376 641C443 641 487 626 509 596C538 555 541 520 550 450L583 450L554 680L33 680L33 641L61 641C96 641 117 639 124 634C131 629 135 617 135 599L135 81C135 63 131 51 124 46C117 41 96 39 61 39L33 39L33 0C59 2 110 3 187 3C274 3 330 2 356 0L356 39L320 39C267 39 238 44 233 55C231 60 230 69 230 82L230 606C230 641 236 641 274 641Z"/></g><g data-mml-node="mo" data-latex="(" transform="translate(625,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234C332-231 330-227 327-223C275-183 233-117 202-26C175 53 161 131 161 208L161 292C161 369 175 447 202 526C233 617 275 683 327 723C330 726 332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 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y="220"/></g><g data-mml-node="mo" data-latex=")" transform="translate(3101.3,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40C268 47 288 129 288 208L288 292C288 371 268 453 229 540C188 631 138 699 78 745C75 747 72 748 71 748C62 748 57 743 57 734C57 730 59 726 62 723C114 683 156 617 187 526C214 447 228 369 228 292L228 208C228 131 214 53 187-26C156-117 114-183 62-223C59-227 57-231 57-234C57-243 62-248 71-248C72-248 75-247 78-245Z"/></g></g></g></svg><mjx-break size="4"> </mjx-break><svg style="vertical-align: -0.781ex;" xmlns="http://www.w3.org/2000/svg" width="7.403ex" height="2.737ex" role="img" focusable="false" viewbox="0 -864.9 3272.3 1209.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\Gamma(\frac{5}{2})=\Gamma(\frac{3}{2}+1)=\frac{3}{2}\Gamma(\frac{3}{2})=\frac{3}{2}\Gamma(\frac{1}{2}+1)=\frac{3}{2}\cdot\frac{1}{2}\Gamma(\frac{1}{2})=\frac{3}{4}\sqrt{\pi}"><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367L80 367C64 367 56 359 56 344C56 329 64 321 80 321L698 321C714 321 722 329 722 344C722 356 711 367 698 367M698 179L80 179C64 179 56 171 56 156C56 141 64 133 80 133L698 133C714 133 722 141 722 156C722 169 711 179 698 179Z"/></g><g data-mml-node="mfrac" data-latex="\frac{3}{4}" transform="translate(1055.8,0)"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)" data-latex="3"><path data-c="33" d="M303 353C369 378 431 441 431 526C431 569 410 604 369 631C333 654 292 666 246 666C201 666 162 654 127 631C88 605 68 571 68 528C68 495 90 472 122 472C154 472 176 495 176 527C176 560 157 578 119 580C145 615 186 633 242 633C302 633 332 598 332 527C332 485 324 450 309 421C282 373 245 364 183 364C171 362 165 357 165 348C165 333 172 333 192 333L235 333C310 333 348 280 348 173C348 88 317 14 241 14C176 14 128 36 99 80C134 79 160 105 160 139C160 173 135 198 101 198C62 198 42 178 42 137C42 88 64 49 108 18C147-9 193-22 244-22C301-22 350-3 393 34C436 71 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transform="translate(853,0)"><g data-mml-node="mi" data-latex="\pi"><path data-c="1D70B" d="M524 431L194 431C153 431 117 415 88 384C74 369 27 305 27 292C31 285 31 279 43 279C50 279 56 283 62 292C94 341 135 366 184 366L235 366C212 279 170 172 110 45C105 33 102 24 102 19C102-1 113-11 134-11C153-11 167 0 176 21C194 78 207 122 214 152L269 366L372 366C345 247 331 164 331 117C331 68 342-11 379-11C400-11 423 9 423 30C423 35 421 43 417 53C398 100 389 153 389 214C389 261 395 312 406 366L515 366C550 366 567 378 567 403C567 426 549 431 524 431Z"/></g></g><rect width="570" height="60" x="853" y="717.5"/></g></g></g></svg></mjx-container></p><p>数学归纳可得：</p><p><mjx-container class="MathJax" jax="SVG" overflow="overflow" display="true" width="full" style="min-width: 22.835ex;"><svg style="vertical-align: -0.566ex; min-width: 22.835ex;" xmlns="http://www.w3.org/2000/svg" width="100%" height="2.262ex" role="img" focusable="false" data-mjx-viewbox="0 -750 10093 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(0.0181,-0.0181) translate(0, -750)"><g data-mml-node="math" data-latex="
\Gamma(n+1)=n!
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332 730 332 734C332 743 327 748 318 748C317 748 314 747 311 745C251 699 201 631 160 540C121 453 101 371 101 292L101 208C101 129 121 47 160-40C201-131 251-199 311-245C314-247 317-248 318-248Z"/></g><g data-mml-node="mi" data-latex="n" transform="translate(1014,0)"><path data-c="1D45B" d="M537 137C514 58 481 18 440 18C427 18 420 28 420 47C420 61 426 84 438 115C478 224 498 296 498 333C498 403 451 442 381 442C322 442 271 416 230 363C222 407 187 442 136 442C80 442 58 390 44 345C34 313 29 294 29 287C29 278 34 273 45 273C50 273 53 274 56 276C61 285 64 292 65 299C83 375 106 413 133 413C151 413 160 399 160 371C160 358 155 331 144 290L87 63C84 50 78 24 78 19C78-1 89-11 111-11C130-11 144-1 151 19C153 24 159 49 170 92L191 181L221 295C232 318 249 341 270 365C299 397 335 413 378 413C411 413 427 391 427 348C427 310 406 234 363 120C356 102 353 87 353 74C353 25 390-11 438-11C482-11 517 14 542 64C561 104 571 131 571 144C571 153 566 158 555 158C552 158 537 149 537 137Z"/></g><g data-mml-node="mo" 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\int_0^{+\infty}x^{n}e^{-x}dx=n!
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