CINXE.COM

Wolfram MathWorld: The Web's Most Extensive Mathematics Resource

<!doctype html> <html lang="en"> <head> <title>Wolfram MathWorld: The Web's Most Extensive Mathematics Resource</title> <meta name="description" content="Comprehensive encyclopedia of mathematics with 13,000 detailed entries. Continually updated, extensively illustrated, and with interactive examples."> <meta property="og:image" content="https://mathworld.wolfram.com/images/socialmedia/share.png"> <meta property="og:url" content="https://mathworld.wolfram.com/"> <meta property="og:type" content="website"> <meta property="og:title" content="Wolfram MathWorld: The Web's Most Extensive Mathematics Resource"> <meta property="og:description" content="Comprehensive encyclopedia of mathematics with 13,000 detailed entries. Continually updated, extensively illustrated, and with interactive examples."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Wolfram MathWorld: The Web's Most Extensive Mathematics Resource"> <meta name="twitter:description" content="Comprehensive encyclopedia of mathematics with 13,000 detailed entries. Continually updated, extensively illustrated, and with interactive examples."> <meta name="twitter:image:src" content="https://mathworld.wolfram.com/images/socialmedia/share.png"> <link rel="canonical" href="https://mathworld.wolfram.com/" /> <meta http-equiv="x-ua-compatible" content="ie=edge"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta charset="utf-8"> <script async src="/common/javascript/analytics.js"></script> <script async src="//www.wolframcdn.com/consent/cookie-consent.js"></script> <script async src="/common/javascript/wal/latest/walLoad.js"></script> <link rel="stylesheet" href="/css/styles.css"> <link rel="preload" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css" as="style" onload="this.onload=null;this.rel='stylesheet'"> <noscript><link rel="stylesheet" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css"></noscript> </head> <body id="home"> <main> <div class="wrapper"> <header class="text-align-c"> <svg id="logo" width="420px" height="68px" viewBox="0 0 420 63" version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g> <g> <g> <polygon fill="#0095AA" points="208.2,47.6 198.9,28.6 192.4,28.6 192.4,64.1 197.1,64.1 197.1,35.3 206.5,54.4 206.5,54.5 209.6,54.5 219.1,35.3 219.1,64.1 223.9,64.1 223.9,28.6 217.5,28.6 "/> <path fill="#0095AA" d="M246.7,38.5c-1.7-1.4-4.5-2.1-8.3-2.1c-2.5,0-4.6,0.4-6.3,1.2c-1.7,0.8-3,2-3.8,3.5l-0.1,0.1l3.8,2.3 l0.1-0.1c0.4-0.7,1.1-1.4,2.1-2c0.9-0.6,2.3-0.9,4.1-0.9c2,0,3.6,0.4,4.7,1.1c1.1,0.7,1.7,2.1,1.7,4.1v1.2l-6.4,0.6 c-2.6,0.3-4.6,0.8-6,1.5c-1.4,0.7-2.5,1.7-3.2,2.9c-0.7,1.2-1.1,2.8-1.1,4.6c0,2.4,0.7,4.5,2.1,6.1c1.4,1.6,3.4,2.4,6,2.4 c1.7,0,3.3-0.4,4.7-1.2c1.3-0.7,2.6-1.7,3.9-2.8v3.2h4.6V45.8C249.2,42.3,248.4,39.9,246.7,38.5z M244.6,50.5v6.3 c-0.5,0.4-1.1,0.9-1.7,1.3c-0.6,0.5-1.2,0.9-1.8,1.2c-0.6,0.4-1.3,0.7-1.9,0.9c-0.6,0.2-1.3,0.4-1.9,0.4c-1.6,0-2.7-0.3-3.5-1 c-0.8-0.7-1.1-1.8-1.1-3.3c0-1.6,0.6-2.9,1.7-3.7c1.2-0.9,3.2-1.4,6.1-1.7L244.6,50.5z"/> <path fill="#0095AA" d="M267.1,60.2c-1.1,0.5-2.3,0.7-3.5,0.7c-1.4,0-2.4-0.4-2.8-1.3c-0.4-0.9-0.6-2.4-0.6-4.6V41h6.8v-3.9h-6.8 v-7.4h-4.6v7.4h-3.8V41h3.8v14.8c0,3.1,0.6,5.4,1.8,6.8c1.2,1.4,3.2,2.2,5.9,2.2c1.7,0,3.4-0.3,5.1-0.8l0.1,0l-1.2-3.9 L267.1,60.2z"/> <path fill="#0095AA" d="M286.1,36.4c-1.7,0-3.4,0.5-5,1.5c-1.5,0.9-2.9,1.9-4.1,3.1v-15h-4.6v38.1h4.6V45.1c1.4-1.2,2.8-2.2,4.3-3 c1.5-0.8,2.8-1.3,3.8-1.3c1.1,0,1.9,0.2,2.5,0.6c0.5,0.4,0.9,1,1.2,1.8c0.2,0.8,0.4,2.5,0.4,5v15.8h4.6V46.1c0-3.5-0.6-6-1.8-7.5 C290.6,37.1,288.7,36.4,286.1,36.4z"/> <polygon fill="#0095AA" points="325.2,56.8 318.6,28.7 318.6,28.6 313,28.6 306.5,56.6 300,28.7 300,28.6 295.1,28.6 303.4,64 303.4,64.1 309.2,64.1 315.7,36.3 322.2,64 322.3,64.1 328.1,64.1 336.6,28.6 331.8,28.6 "/> <path fill="#0095AA" d="M381.5,36.7c-1.7,0-3.2,0.5-4.6,1.5c-1.3,0.9-2.5,1.9-3.6,3.1v-4.1h-4.6v26.9h4.6V45.5 c1.6-1.6,3-2.7,4.2-3.3c1.2-0.6,2.4-0.9,3.7-0.9c0.9,0,1.5,0.1,2,0.3l0.1,0.1l1-4.5l-0.1,0C383.5,36.9,382.6,36.7,381.5,36.7z"/> <rect x="388.4" y="25.9" fill="#0095AA" width="4.6" height="38.1"/> <path fill="#0095AA" d="M415.3,25.9v13.2c-2.2-1.9-4.7-2.8-7.3-2.8c-3.4,0-6.1,1.3-8,3.9c-1.9,2.6-2.9,6.1-2.9,10.5 c0,4.4,0.9,7.9,2.6,10.4c1.8,2.4,4.3,3.7,7.5,3.7c1.6,0,3.1-0.3,4.6-1c1.3-0.6,2.5-1.4,3.4-2.3v2.5h4.7V25.9H415.3z M415.3,43.3 v14c-0.9,0.9-2,1.7-3.2,2.3c-1.2,0.7-2.6,1-3.9,1c-4.1,0-6.2-3.4-6.2-10c0-3.5,0.6-6.1,1.9-7.7c1.2-1.6,2.8-2.5,4.8-2.5 C410.8,40.5,413.1,41.5,415.3,43.3z"/> </g> <g> <polygon fill="#666666" points="41.9,27.6 36.7,27.6 30.3,55.4 23.8,27.8 23.8,27.6 18,27.6 11.6,55.2 5.3,27.6 0,27.6 8.4,63.4 8.4,63.6 14.4,63.6 20.8,36.3 27.2,63.4 27.2,63.6 33.3,63.6 41.8,28 "/> <path fill="#666666" d="M53.4,35.4c-3.9,0-6.9,1.2-8.9,3.6c-2.1,2.4-3.1,6.1-3.1,10.9c0,4.8,1,8.4,3.1,10.8c2.1,2.4,5.1,3.7,9,3.7 c3.9,0,6.9-1.2,8.9-3.7c2-2.4,3.1-6.1,3.1-10.8c0-4.8-1-8.4-3-10.9C60.3,36.6,57.3,35.4,53.4,35.4z M58.6,57.6 c-1.1,1.6-2.8,2.4-5.1,2.4c-2.5,0-4.2-0.8-5.3-2.4C47,56,46.5,53.4,46.5,50c0-3.6,0.6-6.2,1.7-7.8c1.1-1.6,2.9-2.4,5.2-2.4 c2.3,0,4,0.8,5.1,2.4c1.1,1.6,1.7,4.2,1.7,7.8C60.3,53.4,59.7,56,58.6,57.6z"/> <rect x="70.3" y="24.9" fill="#666666" width="5" height="38.7"/> <path fill="#666666" d="M96.7,25.1c-1.6-0.7-3.3-1-5.3-1c-3.1,0-5.4,0.8-6.6,2.4c-1.2,1.6-1.9,4-1.9,7.3v2.3H79v4.3h3.9v23.2h5 V40.5h5.9v-4.3h-5.9v-2.3c0-2.1,0.3-3.6,0.8-4.3c0.5-0.7,1.5-1,2.8-1c0.6,0,1.2,0.1,1.9,0.3c0.7,0.2,1.3,0.4,1.8,0.6l0.3,0.2 l1.4-4.4L96.7,25.1z"/> <path fill="#666666" d="M110.2,35.8c-1.7,0-3.3,0.5-4.6,1.5c-1.2,0.8-2.3,1.8-3.4,2.8v-3.9h-5v27.5h5V44.9 c1.5-1.5,2.9-2.6,4.1-3.2c1.2-0.6,2.4-0.9,3.6-0.9c0.8,0,1.5,0.1,1.9,0.3l0.3,0.1l1.1-4.8l-0.2-0.1 C112.2,35.9,111.3,35.8,110.2,35.8z"/> <path fill="#666666" d="M133.4,37.6c-1.7-1.4-4.5-2.2-8.4-2.2c-2.5,0-4.6,0.4-6.3,1.2c-1.7,0.8-3,2-3.9,3.5l-0.1,0.3l4.1,2.5 l0.2-0.3c0.4-0.7,1.1-1.3,2-1.9c0.9-0.6,2.3-0.9,4-0.9c1.9,0,3.5,0.4,4.6,1.1c1,0.7,1.6,2,1.6,4v1l-6.3,0.6 c-2.6,0.3-4.6,0.8-6,1.6c-1.4,0.7-2.5,1.7-3.2,3c-0.7,1.3-1.1,2.9-1.1,4.8c0,2.5,0.7,4.6,2.1,6.2c1.4,1.6,3.5,2.4,6.1,2.4 c1.8,0,3.4-0.4,4.8-1.2c1.2-0.7,2.4-1.6,3.6-2.6v3h5V45.1C135.9,41.5,135.1,39,133.4,37.6z M123.9,59.9c-1.5,0-2.6-0.3-3.4-1 c-0.7-0.6-1.1-1.7-1.1-3.2c0-1.6,0.5-2.8,1.7-3.6c1.1-0.8,3.1-1.4,6-1.6L131,50v6.1c-0.5,0.4-1.1,0.8-1.6,1.3 c-0.6,0.5-1.2,0.9-1.8,1.2c-0.6,0.4-1.2,0.7-1.9,0.9C125.1,59.8,124.5,59.9,123.9,59.9z"/> <path fill="#666666" d="M176.2,39.6c-0.5-1.4-1.2-2.4-2.3-3.1c-1-0.7-2.4-1.1-4.1-1.1c-1.8,0-3.5,0.5-5,1.6 c-1.3,0.9-2.6,2.1-3.9,3.4c-0.4-1.6-1.1-2.8-2.1-3.6c-1.1-0.9-2.6-1.3-4.5-1.3c-1.7,0-3.3,0.5-4.7,1.5c-1.2,0.8-2.4,1.8-3.4,2.8 v-3.5h-5v27.5h5V44.5c1.3-1.4,2.6-2.4,3.9-3.1c1.2-0.7,2.3-1.1,3.2-1.1c1,0,1.7,0.2,2.1,0.5c0.4,0.3,0.7,0.9,0.9,1.7 c0.2,0.9,0.3,2.6,0.3,5.1v16.1h5V44.5c1.7-1.6,3.1-2.7,4.2-3.3c1-0.6,2-0.9,2.8-0.9c1,0,1.7,0.2,2.1,0.5c0.4,0.3,0.8,0.8,0.9,1.6 c0.2,0.8,0.3,2.6,0.3,5.2v16.1h5V45.4C176.9,42.9,176.7,41,176.2,39.6z"/> </g> <path fill="#0095AA" d="M350.4,34.9c-8.4,0-15.1,6.8-15.1,15.1s6.8,15.1,15.1,15.1c8.4,0,15.1-6.8,15.1-15.1S358.7,34.9,350.4,34.9 z M340.3,57.4c0-0.7,0.1-1.5,0.3-2.3c1.5,1.3,3.4,2.5,5.4,3.4l-0.9,2C343.4,59.7,341.7,58.6,340.3,57.4z M344.8,61.2l-0.3,0.7 c-1.4-0.6-2.6-1.5-3.6-2.3c-0.2-0.3-0.4-0.7-0.5-1.1C341.7,59.5,343.2,60.4,344.8,61.2z M339.6,56.7c-1.2-1.2-2-2.4-2.3-3.5 c-0.2-0.8-0.2-1.6-0.1-2.4c0.6,1.2,1.6,2.5,2.8,3.6C339.8,55.3,339.6,56,339.6,56.7z M339.7,58.3L339.7,58.3 c-0.4-0.5-0.8-1.1-1.1-1.7c0.3,0.4,0.7,0.7,1,1.1C339.6,57.9,339.6,58.1,339.7,58.3z M337.1,48.1c0-0.1,0-0.2,0-0.3 c0.2-1.1,0.5-2.1,1-3.1c0-0.1,0.1-0.1,0.1-0.2c0,0.4,0.1,0.8,0.2,1.2c-0.4,0.6-0.7,1.1-1,1.7C337.2,47.6,337.2,47.8,337.1,48.1z M337.4,49.5c0.1-0.6,0.4-1.1,0.6-1.7c0.2-0.4,0.4-0.8,0.7-1.2c0.6,1.4,1.6,2.7,3,4.1c-0.3,0.6-0.6,1.1-0.9,1.7 c-0.2,0.5-0.4,0.9-0.6,1.3C338.8,52.3,337.9,50.9,337.4,49.5z M362.1,44.9c-0.1,0.6-0.5,1.1-1.1,1.4c-0.5-1.2-1.1-2.3-1.8-3.4 c0.3-0.2,0.5-0.4,0.6-0.7C360.7,43,361.4,43.9,362.1,44.9z M359.8,40.7c0.5,0.3,1,0.7,1.4,1.2c0.4,0.5,0.6,1,0.8,1.5 c0,0.1,0,0.1,0.1,0.2c-0.6-0.8-1.3-1.5-2.1-2.1C359.9,41.3,359.9,41,359.8,40.7C359.8,40.7,359.8,40.7,359.8,40.7z M363.6,48.6 c-0.2,0.9-0.7,1.6-1.5,2.1c-0.1-1.2-0.4-2.4-0.8-3.6c0.6-0.4,1.1-0.8,1.3-1.4C363.1,46.7,363.4,47.6,363.6,48.6z M362.8,44.8 C362.8,44.8,362.9,44.8,362.8,44.8C362.9,44.8,362.9,44.8,362.8,44.8C362.9,44.8,362.9,44.8,362.8,44.8z M345.5,46.7 c0.8-1,1.7-1.9,2.7-2.7c1,0.9,2.2,1.6,3.5,2.3l-1.6,3.4C348.4,48.9,346.8,47.9,345.5,46.7z M349.8,50.5l-1.6,3.5 c-2-0.9-3.9-2.1-5.4-3.5c0.6-1,1.4-2.1,2.2-3.1C346.3,48.6,348,49.6,349.8,50.5z M352.4,46.6c1.1,0.5,2.2,0.8,3.3,1 c0.3,0.1,0.6,0.1,0.9,0.1c0,1.3-0.1,2.5-0.2,3.8c-0.4,0-0.8-0.1-1.2-0.2c-1.4-0.3-2.9-0.7-4.3-1.3 C351.4,48.9,351.9,47.7,352.4,46.6z M356.3,43.4c0.2,1.1,0.3,2.3,0.4,3.5c-0.3,0-0.5-0.1-0.8-0.1c-1-0.2-2.1-0.5-3.1-0.9 c0.5-1.1,0.9-2.1,1.3-3c0.6,0.2,1.1,0.4,1.7,0.5C356,43.3,356.1,43.4,356.3,43.4z M356.2,40.5c0.7,0.6,1.3,1.3,1.8,2 c-0.3,0.1-0.7,0.1-1.2,0.1C356.7,41.8,356.4,41.1,356.2,40.5z M353.8,41.8c-0.5-0.3-1-0.5-1.4-0.9c0.8-0.5,1.6-0.8,2.3-1.1 L353.8,41.8z M353.4,42.5l-1.4,3c-1.2-0.6-2.3-1.3-3.3-2.1c0.9-0.8,1.9-1.5,2.8-2.1C352.2,41.8,352.8,42.2,353.4,42.5z M348.2,42.9c-0.8-0.8-1.4-1.6-1.8-2.4c1.2-0.5,2.5-0.8,3.7-1c0.2,0.4,0.5,0.9,0.9,1.3C350.1,41.4,349.2,42.1,348.2,42.9z M350,38.8c-1.3,0.2-2.5,0.5-3.8,0.9c-0.1-0.7,0-1.4,0.3-1.9c1.1-0.2,2.3-0.2,3.5,0C349.9,38.2,349.9,38.5,350,38.8z M347.6,43.5 c-0.9,0.8-1.8,1.8-2.7,2.7c-1.2-1.2-2.1-2.4-2.5-3.5c1-0.7,2.1-1.4,3.3-1.9C346,41.7,346.7,42.6,347.6,43.5z M345.4,40.1 c-1.2,0.5-2.3,1.1-3.3,1.8c-0.2-1,0-2,0.5-2.7c0.9-0.5,1.9-0.9,2.9-1.1C345.3,38.7,345.2,39.4,345.4,40.1z M344.3,46.9 c-0.8,1-1.6,2-2.2,3.1c-1.4-1.4-2.4-2.8-2.9-4.1c0.7-0.9,1.5-1.8,2.5-2.6C342.2,44.4,343.1,45.7,344.3,46.9z M342.3,51.2 c1.6,1.3,3.4,2.5,5.5,3.5l-1.4,3.1c-2.1-1-3.9-2.2-5.5-3.5c0.2-0.5,0.4-1,0.6-1.6C341.8,52.2,342,51.7,342.3,51.2z M346.9,58.9 c1.7,0.7,3.4,1.3,5.1,1.6c0.4,0.1,0.9,0.2,1.3,0.2c-0.5,0.7-1,1.3-1.6,1.7c-0.5,0-1-0.1-1.4-0.2c-1.4-0.3-2.9-0.7-4.3-1.3 C346.2,60.3,346.5,59.7,346.9,58.9z M352,63.2c0.5,0,0.9,0.1,1.3,0c-0.6,0.1-1.2,0.2-1.9,0.3C351.6,63.4,351.8,63.3,352,63.2z M352.8,62.5c0.5-0.5,1-1,1.4-1.7c1.6,0.1,3.1,0,4.4-0.3c-0.5,0.6-1.1,1-1.8,1.4C355.7,62.3,354.3,62.5,352.8,62.5z M353.7,59.9 c-0.5-0.1-1-0.1-1.6-0.2c-1.7-0.3-3.3-0.8-5-1.5c0.2-0.4,0.4-0.9,0.7-1.4l0.8-1.6c1.7,0.7,3.4,1.3,5.1,1.6c0.5,0.1,1,0.2,1.4,0.2 c-0.2,0.5-0.4,1.1-0.6,1.6C354.3,59,354,59.5,353.7,59.9z M353.9,55.8c-1.6-0.3-3.3-0.8-4.9-1.5c0.5-1.1,1-2.3,1.6-3.5 c1.5,0.6,3,1.1,4.5,1.4c0.4,0.1,0.9,0.1,1.3,0.2c-0.2,1.3-0.5,2.5-0.9,3.7C354.9,56,354.4,55.9,353.9,55.8z M357.1,52.5 c1.7,0.1,3.1-0.1,4.3-0.5c0,1.2-0.1,2.4-0.3,3.5c-1.3,0.6-2.9,0.8-4.8,0.7C356.6,55,356.9,53.8,357.1,52.5z M357.2,51.6 c0.2-1.3,0.2-2.5,0.2-3.8c1.2,0.1,2.3-0.1,3.2-0.4c0.4,1.2,0.7,2.4,0.8,3.7C360.3,51.6,358.8,51.7,357.2,51.6z M360.3,46.6 c-0.8,0.3-1.8,0.4-2.9,0.3c-0.1-1.2-0.2-2.4-0.4-3.5c0.6,0,1.1-0.1,1.5-0.2C359.2,44.3,359.8,45.4,360.3,46.6z M358.7,42.2 c-0.5-0.6-1-1.2-1.5-1.8c0.7,0.4,1.3,0.8,1.9,1.2C359,41.9,358.9,42.1,358.7,42.2z M357.2,39.8c0.5,0.1,1,0.2,1.5,0.4 c0.2,0.2,0.3,0.4,0.3,0.7C358.5,40.4,357.9,40.1,357.2,39.8z M356.3,39c0.3-0.2,0.6-0.3,0.8-0.3c0.3,0.2,0.5,0.4,0.7,0.5 C357.3,39.1,356.8,39,356.3,39z M355.5,38.7c-0.1-0.4-0.1-0.7-0.2-0.9c0.3,0.1,0.7,0.3,1,0.5C356.1,38.3,355.8,38.5,355.5,38.7z M356.1,42.5c-0.1,0-0.1,0-0.2,0c-0.5-0.1-1-0.2-1.5-0.4c0.3-0.7,0.6-1.4,0.9-2C355.6,40.8,355.9,41.6,356.1,42.5z M353.5,37.3 c0.2,0,0.3,0,0.5,0.1c0.1,0,0.3,0.1,0.4,0.1c0.1,0.2,0.2,0.5,0.3,0.8C354.4,37.9,353.9,37.6,353.5,37.3z M353.8,39.4 c-0.7,0.3-1.3,0.6-2,1c-0.3-0.3-0.6-0.7-0.8-1C352,39.3,352.9,39.3,353.8,39.4z M353.2,38.7c-0.8,0-1.6,0-2.4,0 c0-0.3,0-0.5,0.1-0.7C351.7,38.2,352.5,38.4,353.2,38.7z M351.6,37.5c0.2-0.1,0.4-0.1,0.6-0.2c0.4,0.2,0.9,0.5,1.3,0.8 C352.8,37.8,352.2,37.6,351.6,37.5z M350.5,37.3c-0.8-0.1-1.5-0.2-2.2-0.2c-0.3,0-0.5,0-0.8,0c0.5-0.2,1-0.4,1.6-0.5 c0.7,0,1.3,0.1,2,0.3C350.8,37,350.6,37.1,350.5,37.3z M341.4,42.4c-0.9,0.7-1.8,1.5-2.5,2.4c-0.2-0.8-0.1-1.5,0.1-2.1 c0.6-1,1.4-1.9,2.3-2.7C341.2,40.7,341.2,41.6,341.4,42.4z M345.6,61.5c1.5,0.6,3,1.1,4.4,1.4c0.2,0,0.5,0.1,0.7,0.1 c-0.4,0.2-0.8,0.3-1.2,0.3c-0.4,0-0.8-0.1-1.2-0.2c-1-0.2-2.1-0.5-3.1-1C345.4,62.1,345.5,61.8,345.6,61.5z M354.6,60 c0.2-0.4,0.4-0.8,0.7-1.3c0.3-0.6,0.5-1.1,0.7-1.8c1.8,0.1,3.5,0,4.8-0.5c-0.1,0.4-0.3,0.9-0.5,1.3c-0.3,0.6-0.6,1.1-0.9,1.6 C358.2,59.9,356.5,60.1,354.6,60z M361.1,57.9c0.3-0.6,0.5-1.2,0.7-1.9c0.4-0.2,0.7-0.4,1-0.6c-0.4,1-1,1.9-1.6,2.7 c-0.1,0.1-0.2,0.2-0.3,0.3C360.9,58.3,361,58.1,361.1,57.9z M362,55c0.2-1.1,0.3-2.2,0.2-3.4c0.7-0.4,1.3-0.9,1.7-1.5 c0,0.2,0,0.3,0,0.5c0,0.9-0.2,1.8-0.4,2.6C363.2,54,362.7,54.6,362,55z"/> </g> <g> <path fill="#0095AA" d="M2.5,2.9h6.3L8.5,4H3.6L2.8,7.9H7L6.7,9.1H2.6l-1,5.1H0.2L2.5,2.9z"/> <path fill="#0095AA" d="M10.7,2.9h3.1c2.1,0,3.6,0.7,3.6,2.7c0,2.1-1.4,3.4-3.3,3.8l2,4.9h-1.5l-1.9-4.8h-2l-1,4.8H8.4L10.7,2.9z M12.6,8.3c2.2,0,3.3-1,3.3-2.6c0-1.2-0.9-1.7-2.5-1.7h-1.7l-0.8,4.4H12.6z"/> <path fill="#0095AA" d="M18.3,9.7c0-3.5,2.3-7,5.5-7c2.5,0,3.9,1.9,3.9,4.8c0,3.5-2.3,7-5.5,7C19.7,14.4,18.3,12.5,18.3,9.7z M26.4,7.5c0-2.2-0.8-3.7-2.6-3.7c-2.4,0-4.1,3.1-4.1,5.7c0,2.2,0.8,3.7,2.6,3.7C24.7,13.3,26.4,10.2,26.4,7.5z"/> <path fill="#0095AA" d="M31.1,2.9h1.7L33.7,9c0.1,0.8,0.2,1.6,0.3,2.4h0.1c0.4-0.8,0.7-1.6,1.2-2.4l3.3-6.2h1.7l-2.3,11.4h-1.3 l1.3-6.4c0.2-1,0.6-2.4,0.9-3.4h-0.1L37.6,7l-3.3,6h-1l-1-6l-0.3-2.7h-0.1c-0.1,1.1-0.3,2.4-0.5,3.4l-1.3,6.4h-1.2L31.1,2.9z"/> <path fill="#0095AA" d="M48.8,4h-3.3l0.2-1.2h8L53.5,4h-3.3l-2,10.2h-1.4L48.8,4z"/> <path fill="#0095AA" d="M55.6,2.9H57l-1,4.8h5.1l1-4.8h1.4l-2.3,11.4h-1.4l1.1-5.4h-5.1l-1.1,5.4h-1.4L55.6,2.9z"/> <path fill="#0095AA" d="M66.4,2.9h6.3L72.5,4h-4.9l-0.7,3.7H71l-0.2,1.1h-4.2l-0.9,4.3h5.1l-0.2,1.1h-6.4L66.4,2.9z"/> <path fill="#0095AA" d="M78.6,2.9h1.7L81.2,9c0.1,0.8,0.2,1.6,0.3,2.4h0.1c0.4-0.8,0.7-1.6,1.2-2.4l3.3-6.2h1.7l-2.3,11.4h-1.3 l1.3-6.4c0.2-1,0.6-2.4,0.9-3.4h-0.1L85.1,7l-3.3,6h-1l-1-6l-0.3-2.7h-0.1c-0.1,1.1-0.3,2.4-0.5,3.4l-1.3,6.4h-1.2L78.6,2.9z"/> <path fill="#0095AA" d="M93.2,2.9h1.6l1.3,11.4h-1.4l-0.3-3.5h-4l-1.8,3.5h-1.5L93.2,2.9z M90.9,9.6h3.4l-0.2-1.9 C94,6.5,93.9,5.3,93.8,4h-0.1c-0.6,1.3-1.2,2.4-1.8,3.7L90.9,9.6z"/> <path fill="#0095AA" d="M99.7,2.9h1.4l-1.1,5.6h0.1l5.7-5.6h1.5l-4.2,4.4l2.6,7h-1.4l-2.2-6.2l-2.4,2.3l-0.8,3.8h-1.4L99.7,2.9z"/> <path fill="#0095AA" d="M108.6,2.9h6.3L114.7,4h-4.9L109,7.7h4.2L113,8.8h-4.2l-0.9,4.3h5.1l-0.2,1.1h-6.4L108.6,2.9z"/> <path fill="#0095AA" d="M117.4,2.9h3.1c2.1,0,3.6,0.7,3.6,2.7c0,2.1-1.4,3.4-3.3,3.8l2,4.9h-1.5l-1.9-4.8h-2l-1,4.8h-1.4L117.4,2.9 z M119.3,8.3c2.2,0,3.3-1,3.3-2.6c0-1.2-0.9-1.7-2.5-1.7h-1.7l-0.8,4.4H119.3z"/> <path fill="#0095AA" d="M124.4,12.4l0.9-0.8c0.7,1,1.7,1.6,3,1.6c1.4,0,2.4-0.8,2.4-2.1c0-0.9-0.5-1.4-1.4-1.9L128,8.5 c-1-0.6-1.8-1.4-1.8-2.7c0-1.7,1.5-3.2,3.5-3.2c1.4,0,2.5,0.7,3.2,1.6l-0.8,0.8c-0.6-0.7-1.4-1.2-2.5-1.2c-1.3,0-2,0.9-2,1.8 c0,0.9,0.6,1.4,1.5,1.9l1.3,0.8c1.1,0.7,1.7,1.4,1.7,2.7c0,2-1.6,3.5-4,3.5C126.6,14.4,125.3,13.8,124.4,12.4z"/> <path fill="#0095AA" d="M136.7,9.7c0-3.5,2.3-7,5.5-7c2.5,0,3.9,1.9,3.9,4.8c0,3.5-2.3,7-5.5,7C138.1,14.4,136.7,12.5,136.7,9.7z M144.8,7.5c0-2.2-0.8-3.7-2.6-3.7c-2.4,0-4.1,3.1-4.1,5.7c0,2.2,0.8,3.7,2.6,3.7C143.1,13.3,144.8,10.2,144.8,7.5z"/> <path fill="#0095AA" d="M149.5,2.9h6.3L155.6,4h-4.9l-0.8,3.9h4.2l-0.2,1.1h-4.2l-1,5.1h-1.4L149.5,2.9z"/> <path fill="#0095AA" d="M161.4,2.8l2.2,0l0.7,5.7c0.1,0.7,0.1,1.4,0.2,2.2l0.1,0c0.3-0.8,0.6-1.4,1.1-2.2l3.2-5.6l2.3,0l-2.5,11.3 l-1.8,0l1.2-5.4c0.2-1.1,0.7-2.6,1.1-3.7l-0.1,0l-1.4,2.8l-2.9,5.2l-1.3,0l-0.7-5.3L162.4,5l-0.1,0c-0.1,1.1-0.3,2.6-0.5,3.7 l-1.2,5.4l-1.7,0L161.4,2.8z"/> <path fill="#0095AA" d="M176.1,2.9l2.3,0l1,11.4l-2,0l-0.2-3.2l-3.6-0.1l-1.6,3.1l-2.1,0L176.1,2.9z M174.3,9.6l2.8,0l-0.1-1.7 c0-1.1-0.1-2.3-0.1-3.5l-0.1,0c-0.6,1.2-1.1,2.3-1.7,3.5L174.3,9.6z"/> <path fill="#0095AA" d="M184.1,4.5L181,4.5l0.3-1.6l8.2,0.1l-0.4,1.6L186,4.6l-2.1,9.7l-1.9,0L184.1,4.5z"/> <path fill="#0095AA" d="M191,2.8l1.9,0l-1,4.6l4.4,0.1l1-4.6l1.9,0l-2.5,11.3l-1.9,0l1.1-5.1l-4.4-0.1l-1.1,5.1l-1.9,0L191,2.8z"/> <path fill="#0095AA" d="M201.7,2.9l6.5,0.1l-0.4,1.6l-4.6-0.1l-0.6,3l3.9,0.1l-0.4,1.6l-3.9-0.1l-0.8,3.5l4.8,0.1l-0.4,1.6 l-6.7-0.1L201.7,2.9z"/> <path fill="#0095AA" d="M210.2,2.8l2.2,0l0.7,5.7c0.1,0.7,0.1,1.4,0.2,2.2l0.1,0c0.3-0.8,0.6-1.4,1.1-2.2l3.2-5.6l2.3,0l-2.5,11.3 l-1.8,0l1.2-5.4c0.2-1.1,0.7-2.6,1.1-3.7l-0.1,0l-1.4,2.8l-2.9,5.2l-1.3,0l-0.7-5.3L211.2,5l-0.1,0c-0.1,1.1-0.3,2.6-0.5,3.7 l-1.2,5.4l-1.7,0L210.2,2.8z"/> <path fill="#0095AA" d="M224.9,2.9l2.3,0l1,11.4l-2,0l-0.2-3.2l-3.6-0.1l-1.6,3.1l-2.1,0L224.9,2.9z M223.1,9.6l2.8,0l-0.1-1.7 c0-1.1-0.1-2.3-0.1-3.5l-0.1,0c-0.6,1.2-1.1,2.3-1.7,3.5L223.1,9.6z"/> <path fill="#0095AA" d="M232.6,4.5l-3.1-0.1l0.3-1.6L238,3l-0.4,1.6l-3.1-0.1l-2.1,9.7l-1.9,0L232.6,4.5z"/> <path fill="#0095AA" d="M239.5,2.9l1.9,0l-2.5,11.3l-1.9,0L239.5,2.9z"/> <path fill="#0095AA" d="M241.9,9.7c0.1-3.8,2.6-7,6-7c1.3,0,2.3,0.7,2.9,1.7l-1.1,1.1c-0.4-0.7-1.1-1.1-1.9-1.1 c-2.2,0-3.9,2.6-3.9,5.2c0,1.9,0.7,3.2,2.5,3.2c0.9,0,1.6-0.3,2.2-0.9l0.8,1.2c-0.8,0.7-1.9,1.3-3.4,1.3 C243.4,14.4,241.8,12.5,241.9,9.7z"/> <path fill="#0095AA" d="M255.4,2.9l2.3,0l1,11.4l-2,0l-0.2-3.2l-3.6-0.1l-1.6,3.1l-2.1,0L255.4,2.9z M253.6,9.6l2.8,0l-0.1-1.7 c0-1.1-0.1-2.3-0.1-3.5l-0.1,0c-0.6,1.2-1.1,2.3-1.7,3.5L253.6,9.6z"/> <path fill="#0095AA" d="M268.3,2.9h1.6l1.3,11.4h-1.4l-0.3-3.5h-4l-1.8,3.5h-1.5L268.3,2.9z M266,9.6h3.4l-0.2-1.9 c-0.1-1.2-0.2-2.4-0.3-3.7h-0.1c-0.6,1.3-1.2,2.4-1.8,3.7L266,9.6z"/> <path fill="#0095AA" d="M274.5,2.9h1.4l2.5,7.3c0.2,0.5,0.5,1.6,0.7,2.4h0.1c0.1-1.2,0.3-2.4,0.5-3.6l1.2-6.1h1.3L280,14.2h-1.4 L276,6.9c-0.2-0.5-0.5-1.5-0.7-2.4h-0.1c-0.1,1.2-0.3,2.4-0.5,3.6l-1.2,6.1h-1.3L274.5,2.9z"/> <path fill="#0095AA" d="M284.9,2.9h2.1c3.1,0,4.8,1.6,4.8,4.5c0,4.1-2.4,6.9-6.2,6.9h-3.1L284.9,2.9z M285.7,13.1 c2.9,0,4.8-2.4,4.8-5.7c0-2.2-1.3-3.5-3.5-3.5h-1l-1.8,9.2H285.7z"/> <path fill="#0095AA" d="M297.1,2.4l2,0l-0.4,6c-0.1,1.3-0.2,2.5-0.3,3.8l0.1,0c0.5-1.3,1.1-2.6,1.6-3.8l2.6-6l1.9,0l-0.1,6 c0,1.2-0.1,2.5-0.1,3.8l0.1,0c0.5-1.3,1-2.5,1.5-3.8l2.4-6l1.9,0l-4.9,11.6l-2.4,0l0-6.4c0-0.8,0.1-1.8,0.2-2.7l-0.1,0 c-0.3,0.9-0.6,1.8-1,2.7l-2.8,6.3l-2.5,0L297.1,2.4z"/> <path fill="#0095AA" d="M309.7,9.4c0.1-3.9,2.7-7.2,6-7.1c2.7,0,4.2,2,4.2,5c-0.1,3.9-2.7,7.2-6,7.1 C311.2,14.4,309.6,12.5,309.7,9.4z M317.8,7.4c0-2-0.7-3.4-2.3-3.4c-2,0-3.8,2.6-3.8,5.3c0,2,0.7,3.4,2.3,3.4 C316,12.7,317.7,10.1,317.8,7.4z"/> <path fill="#0095AA" d="M322.5,2.5l2,0l-2.2,10l4.7,0.1l-0.4,1.7l-6.7-0.1L322.5,2.5z"/> <path fill="#0095AA" d="M330.8,2.5l6.7,0.1l-0.4,1.7l-4.8-0.1l-0.7,3.4l4,0.1l-0.4,1.7l-4-0.1l-1.1,5l-2,0L330.8,2.5z"/> <path fill="#0095AA" d="M338.8,2.5l3.5,0.1c2.2,0,3.8,0.9,3.8,3c0,2.2-1.4,3.5-3.2,3.9l1.8,4.9l-2.1,0l-1.6-4.6l-1.7,0l-1,4.6l-2,0 L338.8,2.5z M341.1,8c2,0,3-0.9,3-2.3c0-1.1-0.8-1.6-2.2-1.6l-1.5,0L339.6,8L341.1,8z"/> <path fill="#0095AA" d="M351.4,2.5l2.3,0l1.1,11.7l-2.1,0l-0.2-3.3l-3.8-0.1l-1.7,3.2l-2.2,0L351.4,2.5z M349.6,9.4l2.9,0.1 l-0.1-1.8c-0.1-1.2-0.1-2.4-0.2-3.6l-0.1,0c-0.6,1.2-1.1,2.4-1.7,3.6L349.6,9.4z"/> <path fill="#0095AA" d="M358.6,2.4l2.3,0l0.7,5.9c0.1,0.8,0.2,1.5,0.2,2.3l0.1,0c0.4-0.8,0.7-1.5,1.1-2.2l3.3-5.8l2.3,0l-2.5,11.7 l-1.8,0l1.3-5.6c0.3-1.1,0.7-2.7,1.1-3.9l-0.1,0l-1.4,2.9l-3,5.4l-1.4,0l-0.8-5.5l-0.3-2.9l-0.1,0c-0.1,1.1-0.3,2.7-0.5,3.9 l-1.2,5.6l-1.8,0L358.6,2.4z"/> <path fill="#0095AA" d="M371.9,0.1l1.1,0.3l-3.8,16.2l-1.1-0.3L371.9,0.1z"/> <path fill="#0095AA" d="M378.5,2.5l2.3,0l1.1,11.7l-2.1,0l-0.2-3.3l-3.8-0.1l-1.7,3.2l-2.2,0L378.5,2.5z M376.6,9.4l2.9,0.1 l-0.1-1.8c-0.1-1.2-0.1-2.4-0.2-3.6l-0.1,0c-0.6,1.2-1.1,2.4-1.7,3.6L376.6,9.4z"/> <path fill="#0095AA" d="M385.3,2.5l2,0l-2.2,10l4.7,0.1l-0.4,1.7l-6.7-0.1L385.3,2.5z"/> <path fill="#0095AA" d="M393.5,2.5l3.5,0.1c2.2,0,3.8,0.9,3.8,3.1c-0.1,3.1-2.4,4.3-5.3,4.3l-1.6,0l-0.9,4.3l-2,0L393.5,2.5z M395.7,8.3c2,0,3.2-0.8,3.2-2.5c0-1.2-0.8-1.7-2.2-1.7l-1.5,0l-0.9,4.2L395.7,8.3z"/> <path fill="#0095AA" d="M403,2.5l2,0l-1,4.7l4.6,0.1l1-4.7l2,0L409,14.3l-2,0l1.1-5.2l-4.6-0.1l-1.1,5.2l-2,0L403,2.5z"/> <path fill="#0095AA" d="M416.3,2.5l2.3,0l1.1,11.7l-2.1,0l-0.2-3.3l-3.8-0.1l-1.7,3.2l-2.2,0L416.3,2.5z M414.5,9.4l2.9,0.1 l-0.1-1.8c-0.1-1.2-0.1-2.4-0.2-3.6l-0.1,0c-0.6,1.2-1.1,2.4-1.7,3.6L414.5,9.4z"/> </g> </g> <a href="https://www.wolfram.com/mathematica/"><rect x="149.9" y="0" style="fill:#ffffff00;" width="96.7" height="13.2"/></a> <a href="https://wolframalpha.com/"><rect x="297" y="0" style="fill:#ffffff00;" width="121.2" height="13.2"/></a> </svg> <form method="get" action="/search/" name="search" id="search" accept-charset="UTF-8"> <input type="text" name="query" id="searchFieldHome"> <button type="submit"> <img src="/images/header/search-icon.png" width="18" height="18" title="Search" alt="Search"> </button> </form> <div id="slogan"><span>The web's most extensive mathematics resource</span></div> </header> <section id="categories"> <a href="topics/Algebra.html"> <div> <img src="/images/frontpage/algebra.png" width="46" height="46" title="x^2-1" alt="x^2-1"> <h3>Algebra</h3> </div> </a> <a href="topics/FoundationsofMathematics.html"> <div> <img src="/images/frontpage/foundations-math.png" width="46" height="46" title="addition, subtraction, multiplication, division" alt="addition, subtraction, multiplication, division"> <h3>Foundations of Mathematics</h3> </div> </a> <a href="topics/ProbabilityandStatistics.html"> <div> <img src="/images/frontpage/probability.png" width="46" height="46" title="probability curve" alt="probability curve"> <h3>Probability and Statistics</h3> </div> </a> <a href="topics/AppliedMathematics.html"> <div> <img src="/images/frontpage/applied-math.png" width="46" height="46" title="applied math chart" alt="applied math chart"> <h3>Applied Mathematics</h3> </div> </a> <a href="topics/Geometry.html"> <div> <img src="/images/frontpage/geometry.png" width="46" height="46" title="triangle" alt="triangle"> <h3>Geometry</h3> </div> </a> <a href="topics/RecreationalMathematics.html"> <div> <img src="/images/frontpage/recreational-math.png" width="46" height="46" title="cube" alt="cube"> <h3>Recreational Mathematics</h3> </div> </a> <a href="topics/CalculusandAnalysis.html"> <div> <img src="/images/frontpage/calculus.png" width="46" height="46" title="calculus function" alt="calculus function"> <h3>Calculus and Analysis</h3> </div> </a> <a href="topics/HistoryandTerminology.html"> <div> <img src="/images/frontpage/history.png" width="46" height="46" title="Fibonacci spiral" alt="Fibonacci spiral"> <h3>History and Terminology</h3> </div> </a> <a href="topics/Topology.html"> <div> <img src="/images/frontpage/topology.png" width="46" height="46" title="topology knot" alt="topology knot"> <h3>Topology</h3> </div> </a> <a href="topics/DiscreteMathematics.html"> <div> <img src="/images/frontpage/discrete-math.png" width="46" height="46" title="{...}" alt="{...}"> <h3>Discrete Mathematics</h3> </div> </a> <a href="topics/NumberTheory.html"> <div> <img src="/images/frontpage/num-theory.png" width="46" height="46" title="N1" alt="N1"> <h3>Number Theory</h3> </div> </a> <a href="/whatsnew/"> <div> <img src="/images/frontpage/new-in-mw.png" width="46" height="46" title="MathWorld logo" alt="MathWorld logo"> <h3><i>New in MathWorld</i></h3> </div> </a> </section> <section id="text-description" class="text-align-c"> <p class="h2-description"> Created, developed & nurtured by <a href="/about/author.html" class="highlighted">Eric Weisstein</a> with contributions from the world's mathematical community </p> </section> </div> <section id="WAbox"> <div id="WAbox-secondary"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="174" height="26"> <div id="WAbox-tertiary"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions"> <button type="submit" alt="Evaluate on WolframAlpha" title="Evaluate on WolframAlpha"></button> </form> </div> </div> <div id="WAbox-links"> <p class="display-n__320">See also:</p> <ul> <li><a href="https://www.wolframalpha.com/calculators/integral-calculator/" target="_blank">Online Integral Calculator</a></li> <li><a href="https://www.wolframalpha.com/calculators/derivative-calculator/" target="_blank">Online Derivative Calculator</a></li> </ul> </div> </section> </main> <aside id="bottom"> <style> #bottom { padding-bottom: 65px; } #acknowledgment { display:none; } .attribution { font-size: .75rem; font-style: italic; } footer ul li:not(:last-of-type)::after { background: #a3a3a3; margin-left: .3rem; margin-right: .1rem; } @media all and (max-width: 900px) { .attribution { font-size: 12px; } } @media (max-width: 600px) { footer { max-width: 360px; } footer ul { max-width: 360px; } footer ul:nth-child(1) li:nth-child(2):after { content: ""; height: 11px; } footer ul:nth-child(1) li:nth-child(3):after { content: ""; height: 0px; } } </style> <footer> <ul> <li><a href="/about/">About MathWorld</a></li> <li><a href="/classroom/">MathWorld Classroom</a></li> <li><a href="/contact/">Contribute</a></li> <li><a href="https://www.amazon.com/exec/obidos/ASIN/1420072218/ref=nosim/weisstein-20" target="_blank">MathWorld Book</a></li> <li class="display-n display-ib__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> </ul> <ul> <li class="display-n__600"><a href="/whatsnew/">13,208 Entries</a></li> <li class="display-n__600"><a href="/whatsnew/">Last Updated: Thu Nov 21 2024</a></li> <!-- <li><a href="https://www.wolfram.com" target="_blank">&copy;1999&ndash;<span id="copyright-year-end"> Wolfram Research, Inc.</a></li> --> <li><a href="https://www.wolfram.com" target="_blank">&copy;1999&ndash;2024 Wolfram Research, Inc.</a></li> <li><a href="https://www.wolfram.com/legal/terms/mathworld.html" target="_blank">Terms of Use</a></li> </ul> <ul class="wolfram"> <li class="display-n__600 display-n__900"><a href="https://www.wolfram.com" target="_blank" aria-label="Wolfram"><img src="/images/footer/wolfram-logo.png" alt="Wolfram" title="Wolfram" width="121" height="28"></a></li> <li class="display-n__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> <li class="display-n__600"><a href="https://www.wolfram.com/education/" target="_blank">Wolfram for Education</a></li> <li class="attribution">Created, developed and nurtured by Eric Weisstein at&nbsp;Wolfram&nbsp;Research</li> </ul> </footer> </aside> <script type="text/javascript" src="/scripts/scripts.js"></script> <script src="/common/js/c2c/1.0/WolframC2C.js"></script> <script src="/common/js/c2c/1.0/WolframC2CGui.js"></script> <script src="/common/js/c2c/1.0/WolframC2CDefault.js"></script> <link rel="stylesheet" href="/common/js/c2c/1.0/WolframC2CGui.css.en"> <style> .wolfram-c2c-wrapper { padding: 0px !important; border: 0px; } .wolfram-c2c-wrapper:active { border: 0px; } .wolfram-c2c-wrapper:hover { border: 0px; } </style> <script> let c2cWrittings = new WolframC2CDefault({'triggerClass':'mathworld-c2c_above', 'uniqueIdPrefix': 'mathworld-c2c_above-'}); </script> <style> #IPstripe-outer { background: #47a2af; } #IPstripe-outer:hover { background: #0095aa; } </style> <div id="IPstripe-wrap"></div> <script src="/common/stripe/stripe.en.js"></script> </body> </html>

Pages: 1 2 3 4 5 6 7 8 9 10