“List of periodic functions”的意思、由来-开放百科全书

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List of periodic functions

释义

Notes

This is a list of some well-known periodic functions. The constant function {{math|{{var|f}}{{sub| }}({{var|x}}) {{=}} {{var|c}}}}, where {{mvar|c}} is independent of {{mvar|x}}, is periodic with any period, but lacks a fundamental period. A definition is given for some of the following functions, though each function may have many equivalent definitions.

Trigonometric functions

All trigonometric functions listed have period , unless otherwise stated. For the following trigonometric functions:

{{mvar|U_n}} is the {{mvar|n}}th up/down number,

{{mvar|B_n}} is the {{mvar|n}}th Bernoulli number

Name	Symbol	group=nb\|Formulae are given as Taylor series or derived from other entries.	Fourier Series
Sine
cas (mathematics)
Cosine
cis (mathematics)		cos(x) + i sin(x)}}
Tangent			^[1]
Cotangent			{{citation needed>date=March 2019}}
Secant			-
Cosecant			-
Exsecant			-
Excosecant			-
Versine
Vercosine
Coversine
Covercosine
Haversine
Havercosine
Hacoversine
Hacovercosine
Magnitude of sine wave with amplitude, A, and period, T	-		^[2]{{rp>p. 193}}

Sinus-like functions

Trochoid
Cycloid
Clausen function

Non-smooth functions

The following functions take the variable , period and have range to .

The symbol is the floor function of n and is the sign function.

Name	Formula	Fourier Series	Notes
Triangle wave		-	non-continuous first derivative
Sawtooth wave		^[3]	non-continuous
Square wave		-	non-continuous
Cycloid	date=March 2019}}.	-	non-continuous first derivative
Pulse wave	-	-	non-continuous

The following functions are also not smooth:

Tangent (non-continuous)
Cotangent
Secant
Cosecant
Exsecant
Excosecant

Vector-valued functions

Epitrochoid
Epicycloid (special case of the epitrochoid)
Limaçon (special case of the epitrochoid)
Hypotrochoid
Hypocycloid (special case of the hypotrochoid)
Spirograph (special case of the hypotrochoid)

Doubly periodic functions

Jacobi's elliptic functions
Weierstrass's elliptic function

Notes

1. ^ http://web.mit.edu/jorloff/www/18.03-esg/notes/fourier-tan.pdf
2. ^{{cite book | author=Papula, Lothar| title=Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler| publisher=Vieweg+Teubner Verlag | year=2009 | isbn=3834807575}}
3. ^ http://mathworld.wolfram.com/FourierSeries.html

2 : Mathematics-related lists|Types of functions

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