आवर्ती फलनों की सूची

यहाँ प्रमुख आवर्ती फलनों की सूची दी गयी है।

यदि अलग से कुछ नहीं कहा गया है तो यहाँ सूचीबद्ध सभी त्रिकोनमितीय फलनों का आवर्तकाल ${\displaystyle 2\pi }$ समझें। नीचे दिए त्रिकोणमितीय फलनों के लिए,

U_n is the nth up/down number,

B_n is the nth Bernoulli number

नाम	प्रतीक	सूत्र [nb 1]	फुर्ये श्रेणी (Fourier Series)
Sine	${\displaystyle \sin(x)}$	${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}}$	${\displaystyle \sin(x)}$
cas (mathematics)	${\displaystyle \operatorname {cas} (x)}$	${\displaystyle \sin(x)+\cos(x)}$	${\displaystyle \sin(x)+\cos(x)}$
Cosine	${\displaystyle \cos(x)}$	${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}}$	${\displaystyle \cos(x)}$
cis (mathematics)	${\displaystyle e^{ix},\operatorname {cis} (x)}$	cos(x) + i sin(x)	${\displaystyle \cos(x)+i\sin(x)}$
Tangent	${\displaystyle \tan(x)}$	${\displaystyle \sum _{n=0}^{\infty }{\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}}$	${\displaystyle 2\sum _{n=1}^{\infty }(-1)^{n-1}\sin(2nx)}$ [1]
Cotangent	${\displaystyle \cot(x)}$	${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}}$	${\displaystyle i+2i\sum _{n=1}^{\infty }(\cos 2nx-i\sin 2nx)}$ []
Secant	${\displaystyle \sec(x)}$	${\displaystyle \sum _{n=0}^{\infty }{\frac {U_{2n}x^{2n}}{(2n)!}}}$	-
Cosecant	${\displaystyle \csc(x)}$	${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}x^{2n-1}}{(2n)!}}}$	-
Exsecant	${\displaystyle \operatorname {exsec} (x)}$	${\displaystyle \sec(x)-1}$	-
Excosecant	${\displaystyle \operatorname {excsc} (x)}$	${\displaystyle \csc(x)-1}$	-
Versine	${\displaystyle \operatorname {versin} (x)}$	${\displaystyle 1-\cos(x)}$	${\displaystyle 1-\cos(x)}$
Vercosine	${\displaystyle \operatorname {vercosin} (x)}$	${\displaystyle 1+\cos(x)}$	${\displaystyle 1+\cos(x)}$
Coversine	${\displaystyle \operatorname {coversin} (x)}$	${\displaystyle 1-\sin(x)}$	${\displaystyle 1-\sin(x)}$
Covercosine	${\displaystyle \operatorname {covercosin} (x)}$	${\displaystyle 1+\sin(x)}$	${\displaystyle 1+\sin(x)}$
Haversine	${\displaystyle \operatorname {haversin} (x)}$	${\displaystyle {\frac {1-\cos(x)}{2}}}$	${\displaystyle {\frac {1}{2}}-{\frac {1}{2}}\cos(x)}$
Havercosine	${\displaystyle \operatorname {havercosin} (x)}$	${\displaystyle {\frac {1+\cos(x)}{2}}}$	${\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\cos(x)}$
Hacoversine	${\displaystyle \operatorname {hacoversin} (x)}$	${\displaystyle {\frac {1-\sin(x)}{2}}}$	${\displaystyle {\frac {1}{2}}-{\frac {1}{2}}\sin(x)}$
Hacovercosine	${\displaystyle \operatorname {hacovercosin} (x)}$	${\displaystyle {\frac {1+\sin(x)}{2}}}$	${\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\sin(x)}$
Magnitude of sine wave with amplitude, A, and period, T	-	${\displaystyle A|\sin \left({\frac {2\pi }{T}}x\right)|}$	${\displaystyle {\frac {4A}{2\pi }}+\sum _{n\,\mathrm {even} }{\frac {-4A}{\pi }}{\frac {1}{1-n^{2}}}\cos({\frac {2\pi n}{T}}x)}$ [2]:p. 193

The following functions take the variable ${\displaystyle x}$, period ${\displaystyle p}$ and have range ${\displaystyle -1}$ to ${\displaystyle 1}$. The symbol ${\displaystyle \lfloor n\rfloor }$ is the floor function of n and ${\displaystyle \operatorname {sgn} }$ is the sign function.

नाम	सूत्र	फुर्ये श्रेणी	टीका
त्रिभुज तरंग	${\displaystyle {\frac {4}{p}}\left(x-{\frac {p}{2}}\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor }}$	${\displaystyle {\frac {8}{\pi ^{2}}}\sum _{i=0}^{N-1}(-1)^{i}n^{-2}\sin \left({\frac {2\pi nx}{p}}\right)}$	non-continuous first derivative
Sawtooth wave	${\displaystyle 2\left({\frac {x}{p}}-\left\lfloor {\frac {1}{2}}+{\frac {x}{p}}\right\rfloor \right)}$	${\displaystyle {\frac {4}{\pi }}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {1}{n}}\sin \left({\frac {n\pi x}{p/2}}\right)}$ [3]	non-continuous
वर्ग तरंग	${\displaystyle \operatorname {sgn} \left(\sin {\frac {2\pi x}{p}}\right)}$	-	non-continuous
Cycloid	No closed form	-	non-continuous first derivative
Pulse wave	-	-	non-continuous

The following functions are also not smooth:

• Tangent (non-continuous)
• Cotangent
• Secant
• Cosecant
• Exsecant
• Excosecant

• 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)

• Jacobi's elliptic functions
• Weierstrass's elliptic function

1. ↑ "संग्रहीत प्रति" (PDF). मूल से 31 मार्च 2019 को पुरालेखित (PDF). अभिगमन तिथि 16 अप्रैल 2020.
2. ↑ Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler. Vieweg+Teubner Verlag. आई॰ऍस॰बी॰ऍन॰ 3834807575.
3. ↑ "संग्रहीत प्रति". मूल से 18 मार्च 2020 को पुरालेखित. अभिगमन तिथि 16 अप्रैल 2020.