{"id":3221,"date":"2018-10-18T04:05:55","date_gmt":"2018-10-18T04:05:55","guid":{"rendered":"https:\/\/physicscatalyst.com\/graduation\/?p=3221"},"modified":"2021-06-05T13:45:32","modified_gmt":"2021-06-05T13:45:32","slug":"fourier-series-formula-sheet","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/graduation\/fourier-series-formula-sheet\/","title":{"rendered":"Fourier Series formula sheet"},"content":{"rendered":"

Fourier series is an expansion of a periodic function of period $2\\pi$ which is representation of a function in a series of sine or cosine such as<\/p>\n

$f(x)=a_{0}+\\sum_{n=1}^{\\infty }a_{n}cos(nx)+\\sum_{n=1}^{\\infty }b_{n}sin(nx)$<\/p>\n

where $a_{0}$ , $a_{n}$ and $b_{n}$ are constants and are known as fourier <\/strong>coefficients<\/strong>.
\nIn applying fourier theorem for analysis of an complex periodic function , given function must satisfy following condition
\n(i) It should be single valued
\n(ii) It should be continuous.<\/p>\n

 Drichlet’s Conditions(sufficient but not necessary)<\/strong><\/em><\/p>\n

When a function $f(x)$ is to be expanded in the interval (a,b)
\n(a) $f(a)$ is continous in interval (a,b) except for finite number of finite discontinuties.
\n(b) $f(x)$ has finite number of maxima and minima in this interval.<\/p>\n

Orthogonal property of sine and cosine functions<\/strong>
\n$\\int_{-\\pi}^{\\pi}sin(mx)cos(mx)dx=0$
\n$\\int_{-\\pi}^{\\pi}sin(mx)sin(nx)dx= \\int_{-\\pi}^{\\pi}sin(mx)sin(nx)dx=\\begin{bmatrix}
\n\\pi\\delta_{mn} &m\\neq 0 \\\\
\n0& m=0
\n\\end{bmatrix} $
\n$\\int_{-\\pi}^{\\pi}cos(mx)cos(nx)dx=\\begin{bmatrix}
\n\\pi\\delta_{mn} &m\\neq 0 \\\\
\n2\\pi& m=0
\n\\end{bmatrix} $
\nFourier Constants<\/strong>
\n$a_{0}=\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi}f(x)dx$
\n$a_{0}$  is the average value of function $f(x)$ over the interval
\n$a_{n}=\\frac{1}{\\pi}\\int_{-\\pi}^{\\pi}f(x)cos(nx)dx$
\n$b_{n}=\\frac{1}{\\pi}\\int_{-\\pi}^{\\pi}f(x)sin(nx)dx$
\nFor even functions<\/strong>
\n$f(-x)=f(x)$ and fourier series becomes
\n$f(x)=a_{0}+\\sum_{n=1}^{\\infty }a_{n}cos(nx)$
\n For odd functions<\/strong>
\n$f(-x)=-f(x)$ and fourier series becomes
\n$f(x)=a_{0}+\\sum_{n=1}^{\\infty }a_{n}sin(nx)$
\nComplex form of fourier series<\/strong>
\nputting $c_{0}=c_{0}$
\n$c_{n}=\\frac{a_{n}-ib_{n}}{2}$
\nand
\n$c_{-n}=\\frac{a_{n}+ib_{n}}{2}$
\n$f(x)=\\sum_{-\\infty }^{\\infty }C_{n}e^{inx}$
\ncoefficent
\n$C_{n}=\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi}f(x)e^{-inx}dx$
\nFourier series in interval (0,T)<\/strong>
\nGeneral fourier series of a periodic piecewise continous function $f(T)$ having period $T=\\frac{2\\pi}{\\omega}$ is
\n$f(t)=a_{0}+\\sum_{n=1}^{\\infty }a_{n}cos(nx)+\\sum_{n=1}^{\\infty }b_{n}sin(nx)$
\nwhere
\n$a_{0}=\\frac{1}{T}\\int_{0}^{T}f(t)dt$
\n$a_{n}=\\frac{2}{T}\\int_{0}^{T}f(t)cos(n\\omega T)dt$
\n$b_{n}=\\frac{2}{T}\\int_{0}^{T}f(t)sin(n\\omega T)dt$
\n Complex Form of Fourier Series<\/strong>
\n$f(x)=\\sum_{n=-\\infty }^{\\infty }C_{n}e^{-i\\omega t}$
\nwhere
\n$c_{n}=\\frac{1}{T}\\int_{0}^{T}f(t)e^{-i\\omega t}dx$
\n Advantages of Fourier series<\/strong>
\n1. It can also represent discontinous functions
\n2.  Even and odd functions are conveniently represented as cosine and sine series.
\n3.  Fourier expansion gives no assurance of its validity outside the interval.<\/p>\n

Change of interval from<\/strong> $(-\\pi,\\pi)$ to $(-l,l)$
\nSeries will be
\n$f(x)=a_{0}+\\sum_{n=1}^{\\infty }a_{n}cos(\\frac{nx\\pi}{l})+\\sum_{n=1}^{\\infty }b_{n}sin(\\frac{nx\\pi}{l})$
\nwith
\n$a_{0}=\\frac{1}{2l}\\int_{-l}^{l}f(x)dx$
\n$a_{n}=\\frac{1}{2l}\\int_{-l}^{l}f(x)cos(\\frac{n\\pi x}{l})dx$
\n$b_{n}=\\frac{1}{2l}\\int_{-l}^{l}f(x)sin(\\frac{n\\pi x}{l})dx$
\nFourier Series in interval<\/strong> $(0,l)$
\nCosine series when function $f(x)$ is even
\n$f(x)=a_{0}+\\sum_{n=1}^{\\infty }a_{n}cos(\\frac{n\\pi x}{l})$<\/p>\n

Download Fourier Series formula sheet as PDF<\/a><\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"

Fourier series is an expansion of a periodic function of period $2\\pi$ which is representation of a function in a series of sine or cosine such as $f(x)=a_{0}+\\sum_{n=1}^{\\infty }a_{n}cos(nx)+\\sum_{n=1}^{\\infty }b_{n}sin(nx)$ where $a_{0}$ , $a_{n}$ and $b_{n}$ are constants and are known as fourier coefficients. In applying fourier theorem for analysis of an complex periodic function , given function must satisfy following condition (i) It […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[49],"tags":[],"class_list":["post-3221","post","type-post","status-publish","format-standard","hentry","category-mathematical-physics"],"yoast_head":"\nFourier Series formula sheet - Learn about education and B.Sc. 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