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List of periodic functions
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<h2 id="smooth-functions">Smooth functions</h2> <p>All trigonometric functions listed have period 2 π {\displaystyle 2\pi } , unless otherwise stated. For the following trigonometric functions: </p> Un is the nth <a href="/facts/Up%2fdown_number/Fl7tSe9z">up/down number</a>, Bn is the nth <a href="/facts/Bernoulli_number/RH0mblhs">Bernoulli number</a> in Jacobi elliptic functions, q = e − π K ( 1 − m ) K ( m ) {\displaystyle q=e^{-\pi {\frac {K(1-m)}{K(m)}}}} <table><tbody><tr><th>Name</th><th>Symbol</th><th>Formula <a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></th><th>Fourier Series</th></tr><tr><td><a href="/facts/Sine/gQ6LXK8z">Sine</a></td><td> sin ( x ) {\displaystyle \sin(x)} </td><td> ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}} </td><td> sin ( x ) {\displaystyle \sin(x)} </td></tr><tr><td><a href="/facts/Cas_(mathematics)/9p2gPkmU">cas (mathematics)</a></td><td> cas ( x ) {\displaystyle \operatorname {cas} (x)} </td><td> sin ( x ) + cos ( x ) {\displaystyle \sin(x)+\cos(x)} </td><td> sin ( x ) + cos ( x ) {\displaystyle \sin(x)+\cos(x)} </td></tr><tr><td><a href="/facts/Cosine/czej7ur0">Cosine</a></td><td> cos ( x ) {\displaystyle \cos(x)} </td><td> ∑ n = 0 ∞ ( − 1 ) n x 2 n ( 2 n ) ! {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}} </td><td> cos ( x ) {\displaystyle \cos(x)} </td></tr><tr><td><a href="/facts/Cis_(mathematics)/1JA8Tsgb">cis (mathematics)</a></td><td> e i x , cis ( x ) {\displaystyle e^{ix},\operatorname {cis} (x)} </td><td> cos(<i>x</i>) + <i>i</i> sin(<i>x</i>)</td><td> cos ( x ) + i sin ( x ) {\displaystyle \cos(x)+i\sin(x)} </td></tr><tr><td><a href="/facts/Tangent_(trigonometry)/czej7ur0">Tangent</a></td><td> tan ( x ) {\displaystyle \tan(x)} </td><td> sin x cos x = ∑ n = 0 ∞ U 2 n + 1 x 2 n + 1 ( 2 n + 1 ) ! {\displaystyle {\frac {\sin x}{\cos x}}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}} </td><td> 2 ∑ n = 1 ∞ ( − 1 ) n − 1 sin ( 2 n x ) {\displaystyle 2\sum _{n=1}^{\infty }(-1)^{n-1}\sin(2nx)} <a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></td></tr><tr><td><a href="/facts/Cotangent/czej7ur0">Cotangent</a></td><td> cot ( x ) {\displaystyle \cot(x)} </td><td> cos x sin x = ∑ n = 0 ∞ ( − 1 ) n 2 2 n B 2 n x 2 n − 1 ( 2 n ) ! {\displaystyle {\frac {\cos x}{\sin x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}} </td><td> i + 2 i ∑ n = 1 ∞ ( cos 2 n x − i sin 2 n x ) {\displaystyle i+2i\sum _{n=1}^{\infty }(\cos 2nx-i\sin 2nx)} </td></tr><tr><td><a href="/facts/Secant_(trigonometry)/czej7ur0">Secant</a></td><td> sec ( x ) {\displaystyle \sec(x)} </td><td> 1 cos x = ∑ n = 0 ∞ U 2 n x 2 n ( 2 n ) ! {\displaystyle {\frac {1}{\cos x}}=\sum _{n=0}^{\infty }{\frac {U_{2n}x^{2n}}{(2n)!}}} </td><td>-</td></tr><tr><td><a href="/facts/Cosecant/czej7ur0">Cosecant</a></td><td> csc ( x ) {\displaystyle \csc(x)} </td><td> 1 sin x = ∑ n = 0 ∞ ( − 1 ) n + 1 2 ( 2 2 n − 1 − 1 ) B 2 n x 2 n − 1 ( 2 n ) ! {\displaystyle {\frac {1}{\sin x}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}x^{2n-1}}{(2n)!}}} </td><td>-</td></tr><tr><td><a href="/facts/Exsecant/CNKrYMjC">Exsecant</a></td><td> exsec ( x ) {\displaystyle \operatorname {exsec} (x)} </td><td> sec ( x ) − 1 {\displaystyle \sec(x)-1} </td><td>-</td></tr><tr><td><a href="/facts/Excosecant/CNKrYMjC">Excosecant</a></td><td> excsc ( x ) {\displaystyle \operatorname {excsc} (x)} </td><td> csc ( x ) − 1 {\displaystyle \csc(x)-1} </td><td>-</td></tr><tr><td><a href="/facts/Versine/MllE4ISd">Versine</a></td><td> versin ( x ) {\displaystyle \operatorname {versin} (x)} </td><td> 1 − cos ( x ) {\displaystyle 1-\cos(x)} </td><td> 1 − cos ( x ) {\displaystyle 1-\cos(x)} </td></tr><tr><td><a href="/facts/Vercosine/MllE4ISd">Vercosine</a></td><td> vercosin ( x ) {\displaystyle \operatorname {vercosin} (x)} </td><td> 1 + cos ( x ) {\displaystyle 1+\cos(x)} </td><td> 1 + cos ( x ) {\displaystyle 1+\cos(x)} </td></tr><tr><td><a href="/facts/Coversine/MllE4ISd">Coversine</a></td><td> coversin ( x ) {\displaystyle \operatorname {coversin} (x)} </td><td> 1 − sin ( x ) {\displaystyle 1-\sin(x)} </td><td> 1 − sin ( x ) {\displaystyle 1-\sin(x)} </td></tr><tr><td><a href="/facts/Covercosine/MllE4ISd">Covercosine</a></td><td> covercosin ( x ) {\displaystyle \operatorname {covercosin} (x)} </td><td> 1 + sin ( x ) {\displaystyle 1+\sin(x)} </td><td> 1 + sin ( x ) {\displaystyle 1+\sin(x)} </td></tr><tr><td><a href="/facts/Haversine/MllE4ISd">Haversine</a></td><td> haversin ( x ) {\displaystyle \operatorname {haversin} (x)} </td><td> 1 − cos ( x ) 2 {\displaystyle {\frac {1-\cos(x)}{2}}} </td><td> 1 2 − 1 2 cos ( x ) {\displaystyle {\frac {1}{2}}-{\frac {1}{2}}\cos(x)} </td></tr><tr><td><a href="/facts/Havercosine/MllE4ISd">Havercosine</a></td><td> havercosin ( x ) {\displaystyle \operatorname {havercosin} (x)} </td><td> 1 + cos ( x ) 2 {\displaystyle {\frac {1+\cos(x)}{2}}} </td><td> 1 2 + 1 2 cos ( x ) {\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\cos(x)} </td></tr><tr><td><a href="/facts/Hacoversine/MllE4ISd">Hacoversine</a></td><td> hacoversin ( x ) {\displaystyle \operatorname {hacoversin} (x)} </td><td> 1 − sin ( x ) 2 {\displaystyle {\frac {1-\sin(x)}{2}}} </td><td> 1 2 − 1 2 sin ( x ) {\displaystyle {\frac {1}{2}}-{\frac {1}{2}}\sin(x)} </td></tr><tr><td><a href="/facts/Hacovercosine/MllE4ISd">Hacovercosine</a></td><td> hacovercosin ( x ) {\displaystyle \operatorname {hacovercosin} (x)} </td><td> 1 + sin ( x ) 2 {\displaystyle {\frac {1+\sin(x)}{2}}} </td><td> 1 2 + 1 2 sin ( x ) {\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\sin(x)} </td></tr><tr><td><a href="/facts/Jacobi_elliptic_function/BLk6Rxew">Jacobi elliptic function</a> sn</td><td> sn ( x , m ) {\displaystyle \operatorname {sn} (x,m)} </td><td> sin am ( x , m ) {\displaystyle \sin \operatorname {am} (x,m)} </td><td> 2 π K ( m ) m ∑ n = 0 ∞ q n + 1 / 2 1 − q 2 n + 1 sin ( 2 n + 1 ) π x 2 K ( m ) {\displaystyle {\frac {2\pi }{K(m){\sqrt {m}}}}\sum _{n=0}^{\infty }{\frac {q^{n+1/2}}{1-q^{2n+1}}}~\sin {\frac {(2n+1)\pi x}{2K(m)}}} </td></tr><tr><td><a href="/facts/Jacobi_elliptic_function/BLk6Rxew">Jacobi elliptic function</a> cn</td><td> cn ( x , m ) {\displaystyle \operatorname {cn} (x,m)} </td><td> cos am ( x , m ) {\displaystyle \cos \operatorname {am} (x,m)} </td><td> 2 π K ( m ) m ∑ n = 0 ∞ q n + 1 / 2 1 + q 2 n + 1 cos ( 2 n + 1 ) π x 2 K ( m ) {\displaystyle {\frac {2\pi }{K(m){\sqrt {m}}}}\sum _{n=0}^{\infty }{\frac {q^{n+1/2}}{1+q^{2n+1}}}~\cos {\frac {(2n+1)\pi x}{2K(m)}}} </td></tr><tr><td><a href="/facts/Jacobi_elliptic_function/BLk6Rxew">Jacobi elliptic function</a> dn</td><td> dn ( x , m ) {\displaystyle \operatorname {dn} (x,m)} </td><td> 1 − m sn 2 ( x , m ) {\displaystyle {\sqrt {1-m\operatorname {sn} ^{2}(x,m)}}} </td><td> π 2 K ( m ) + 2 π K ( m ) ∑ n = 1 ∞ q n 1 + q 2 n cos n π x K ( m ) {\displaystyle {\frac {\pi }{2K(m)}}+{\frac {2\pi }{K(m)}}\sum _{n=1}^{\infty }{\frac {q^{n}}{1+q^{2n}}}~\cos {\frac {n\pi x}{K(m)}}} </td></tr><tr><td><a href="/facts/Jacobi_elliptic_function/BLk6Rxew">Jacobi elliptic function</a> zn</td><td> zn ( x , m ) {\displaystyle \operatorname {zn} (x,m)} </td><td> ∫ 0 x [ dn ( t , m ) 2 − E ( m ) K ( m ) ] d t {\displaystyle \int _{0}^{x}\left[\operatorname {dn} (t,m)^{2}-{\frac {E(m)}{K(m)}}\right]dt} </td><td> 2 π K ( m ) ∑ n = 1 ∞ q n 1 − q 2 n sin n π x K ( m ) {\displaystyle {\frac {2\pi }{K(m)}}\sum _{n=1}^{\infty }{\frac {q^{n}}{1-q^{2n}}}~\sin {\frac {n\pi x}{K(m)}}} </td></tr><tr><td><a href="/facts/Weierstrass_elliptic_function/GHRXDknk">Weierstrass elliptic function</a></td><td> ℘ ( x , Λ ) {\displaystyle \wp (x,\Lambda )} </td><td> 1 x 2 + ∑ λ ∈ Λ − { 0 } [ 1 ( x − λ ) 2 − 1 λ 2 ] {\displaystyle {\frac {1}{x^{2}}}+\sum _{\lambda \in \Lambda -\{0\}}\left[{\frac {1}{(x-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right]} </td><td> {\displaystyle } </td></tr><tr><td><a href="/facts/Clausen_function/1mDkvwcL">Clausen function</a></td><td> Cl 2 ( x ) {\displaystyle \operatorname {Cl} _{2}(x)} </td><td> − ∫ 0 x ln | 2 sin t 2 | d t {\displaystyle -\int _{0}^{x}\ln \left|2\sin {\frac {t}{2}}\right|dt} </td><td> ∑ k = 1 ∞ sin k x k 2 {\displaystyle \sum _{k=1}^{\infty }{\frac {\sin kx}{k^{2}}}} </td></tr></tbody></table> <h2 id="non-smooth-functions">Non-smooth functions</h2> <p>The following functions have period p {\displaystyle p} and take x {\displaystyle x} as their argument. The symbol ⌊ n ⌋ {\displaystyle \lfloor n\rfloor } is the <a href="/facts/Floor_and_ceiling_functions/ssehyMMf">floor function</a> of n {\displaystyle n} and sgn {\displaystyle \operatorname {sgn} } is the <a href="/facts/Sign_function/VADy9zQq">sign function</a>. </p><p> K means <a href="/facts/Elliptic_integral/q8EGYzQ3">Elliptic integral</a> K(m) </p> <table><tbody><tr><th>Name</th><th>Formula</th><th>Limit</th><th>Fourier Series</th><th>Notes</th></tr><tr><td><a href="/facts/Triangle_wave/72BUtbRl">Triangle wave</a></td><td> 4 p ( x − p 2 ⌊ 2 x p + 1 2 ⌋ ) ( − 1 ) ⌊ 2 x p + 1 2 ⌋ {\displaystyle {\frac {4}{p}}\left(x-{\frac {p}{2}}\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor }} </td><td> lim m → 1 − zs ( 4 K x p − K , m ) {\displaystyle \lim _{m\rightarrow 1^{-}}\operatorname {zs} \left({\frac {4Kx}{p}}-K,m\right)} </td><td> 8 π 2 ∑ n o d d ∞ ( − 1 ) ( n − 1 ) / 2 n 2 sin ( 2 π n x p ) {\displaystyle {\frac {8}{\pi ^{2}}}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {(-1)^{(n-1)/2}}{n^{2}}}\sin \left({\frac {2\pi nx}{p}}\right)} </td><td>non-continuous first derivative</td></tr><tr><td><a href="/facts/Sawtooth_wave/3QA8wW0h">Sawtooth wave</a></td><td> 2 ( x p − ⌊ 1 2 + x p ⌋ ) {\displaystyle 2\left({\frac {x}{p}}-\left\lfloor {\frac {1}{2}}+{\frac {x}{p}}\right\rfloor \right)} </td><td> − lim m → 1 − zn ( 2 K x p + K , m ) {\displaystyle -\lim _{m\rightarrow 1^{-}}\operatorname {zn} \left({\frac {2Kx}{p}}+K,m\right)} </td><td> 2 π ∑ n = 1 ∞ ( − 1 ) n − 1 n sin ( 2 π n x p ) {\displaystyle {\frac {2}{\pi }}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}\sin \left({\frac {2\pi nx}{p}}\right)} </td><td>non-continuous</td></tr><tr><td><a href="/facts/Square_wave_(waveform)/h1cWR1KM">Square wave</a></td><td> sgn ( sin 2 π x p ) {\displaystyle \operatorname {sgn} \left(\sin {\frac {2\pi x}{p}}\right)} </td><td> lim m → 1 − sn ( 4 K x p , m ) {\displaystyle \lim _{m\rightarrow 1^{-}}\operatorname {sn} \left({\frac {4Kx}{p}},m\right)} </td><td> 4 π ∑ n o d d ∞ 1 n sin ( 2 π n x p ) {\displaystyle {\frac {4}{\pi }}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {1}{n}}\sin \left({\frac {2\pi nx}{p}}\right)} </td><td>non-continuous</td></tr><tr><td><a href="/facts/Pulse_wave/CEPen7fY">Pulse wave</a></td><td> H ( cos 2 π x p − cos π t p ) {\displaystyle H\left(\cos {\frac {2\pi x}{p}}-\cos {\frac {\pi t}{p}}\right)} <p>where H {\displaystyle H} is the <a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a>t is how long the pulse stays at 1</p></td><td></td><td> t p + ∑ n = 1 ∞ 2 n π sin ( π n t p ) cos ( 2 π n x p ) {\displaystyle {\frac {t}{p}}+\sum _{n=1}^{\infty }{\frac {2}{n\pi }}\sin \left({\frac {\pi nt}{p}}\right)\cos \left({\frac {2\pi nx}{p}}\right)} </td><td>non-continuous</td></tr><tr><td>Magnitude of sine wave with amplitude, A, and period, p/2</td><td> A | sin π x p | {\displaystyle A\left|\sin {\frac {\pi x}{p}}\right|} </td><td></td><td> 4 A 2 π + ∑ n = 1 ∞ 4 A π 1 4 n 2 − 1 cos 2 π n x p {\displaystyle {\frac {4A}{2\pi }}+\sum _{n=1}^{\infty }{\frac {4A}{\pi }}{\frac {1}{4n^{2}-1}}\cos {\frac {2\pi nx}{p}}} <a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>: p. 193 </td><td>non-continuous</td></tr><tr><td><a href="/facts/Cycloid/8Qokqud5">Cycloid</a></td><td> p − p cos ( f ( − 1 ) ( 2 π x p ) ) 2 π {\displaystyle {\frac {p-p\cos \left(f^{(-1)}\left({\frac {2\pi x}{p}}\right)\right)}{2\pi }}} <p>given f ( x ) = x − sin ( x ) {\displaystyle f(x)=x-\sin(x)} and f ( − 1 ) ( x ) {\displaystyle f^{(-1)}(x)} is</p><p>its real-valued inverse.</p></td><td></td><td> p π ( 3 4 + ∑ n = 1 ∞ J n ( n ) − J n − 1 ( n ) n cos 2 π n x p ) {\displaystyle {\frac {p}{\pi }}{\biggl (}{\frac {3}{4}}+\sum _{n=1}^{\infty }{\frac {\operatorname {J} _{n}(n)-\operatorname {J} _{n-1}(n)}{n}}\cos {\frac {2\pi nx}{p}}{\biggr )}} <p>where J n ( x ) {\displaystyle \operatorname {J} _{n}(x)} is the <a href="/facts/Bessel_function/hSyvFPqC">Bessel Function of the first kind</a>.</p></td><td>non-continuous first derivative</td></tr><tr><td><a href="/facts/Dirac_comb/0tx3xslS">Dirac comb</a></td><td> ∑ n = − ∞ ∞ δ ( x − n p ) {\displaystyle \sum _{n=-\infty }^{\infty }\delta (x-np)} </td><td> lim m → 1 − 2 K ( m ) p π dn ( 2 K x p , m ) {\displaystyle \lim _{m\rightarrow 1^{-}}{\frac {2K(m)}{p\pi }}\operatorname {dn} \left({\frac {2Kx}{p}},m\right)} </td><td> 1 p ∑ n = − ∞ ∞ e 2 n π i x p {\displaystyle {\frac {1}{p}}\sum _{n=-\infty }^{\infty }e^{\frac {2n\pi ix}{p}}} </td><td>non-continuous</td></tr><tr><td><a href="/facts/Dirichlet_function/d9o708HR">Dirichlet function</a></td><td> 1 Q ( x ) = { 1 x ∈ Q 0 x ∉ Q {\displaystyle {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)={\begin{cases}1&x\in \mathbb {Q} \\0&x\notin \mathbb {Q} \end{cases}}}} </td><td> lim m , n → ∞ cos 2 m ( n ! x π ) {\displaystyle \lim _{m,n\rightarrow \infty }\cos ^{2m}(n!x\pi )} </td><td>-</td><td>non-continuous</td></tr></tbody></table> <h2 id="vector-valued-functions">Vector-valued functions</h2> <ul><li><a href="/facts/Epitrochoid/Vdl8VHfH">Epitrochoid</a></li> <li><a href="/facts/Epicycloid/7XIdc9sw">Epicycloid</a> (special case of the epitrochoid)</li> <li><a href="/facts/Lima%25C3%25A7on/bKcmDTM1">Limaçon</a> (special case of the epitrochoid)</li> <li><a href="/facts/Hypotrochoid/B2vtvznY">Hypotrochoid</a></li> <li><a href="/facts/Hypocycloid/XH6rx5I0">Hypocycloid</a> (special case of the hypotrochoid)</li> <li><a href="/facts/Spirograph/c6p3lTSo">Spirograph</a> (special case of the hypotrochoid)</li></ul> <h2 id="doubly-periodic-functions">Doubly periodic functions</h2> <ul><li><a href="/facts/Jacobi%2527s_elliptic_functions/BLk6Rxew">Jacobi's elliptic functions</a></li> <li><a href="/facts/Weierstrass%2527s_elliptic_function/GHRXDknk">Weierstrass's elliptic function</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Formulae are given as Taylor series or derived from other entries. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Orloff, Jeremy. "ES.1803 Fourier Expansion of tan(x)" (PDF). Massachusetts Institute of Technology. Archived from the original (PDF) on 2019-03-31. <a href="https://web.archive.org/web/20190331130103/http://web.mit.edu/jorloff/www/18.03-esg/notes/fourier-tan.pdf" target="_blank">https://web.archive.org/web/20190331130103/http://web.mit.edu/jorloff/www/18.03-esg/notes/fourier-tan.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag. ISBN 978-3834807571. <a href="978-3834807571" target="_blank">978-3834807571</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>