आवर्ती फलनों की सूची

यहाँ प्रमुख आवर्ती फलनों की सूची दी गयी है।

त्रिकोणमितीय फलन

यदि अलग से कुछ नहीं कहा गया है तो यहाँ सूचीबद्ध सभी त्रिकोनमितीय फलनों का आवर्तकाल $2\pi$ 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	फुर्ये श्रेणी (Fourier Series)
Sine	$\sin(x)$	$\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}$	$\sin(x)$
cas (mathematics)	$\operatorname {cas} (x)$	$\sin(x)+\cos(x)$	$\sin(x)+\cos(x)$
Cosine	$\cos(x)$	$\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}$	$\cos(x)$
cis (mathematics)	$e^{ix},\operatorname {cis} (x)$	cos(x) + i sin(x)	$\cos(x)+i\sin(x)$
Tangent	$\tan(x)$	$\sum _{n=0}^{\infty }{\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}$	$2\sum _{n=1}^{\infty }(-1)^{n-1}\sin(2nx)$
Cotangent	$\cot(x)$	$\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}$	$i+2i\sum _{n=1}^{\infty }(\cos 2nx-i\sin 2nx)$ [उद्धरण चाहिए]
Secant	$\sec(x)$	$\sum _{n=0}^{\infty }{\frac {U_{2n}x^{2n}}{(2n)!}}$	-
Cosecant	$\csc(x)$	$\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}x^{2n-1}}{(2n)!}}$	-
Exsecant	$\operatorname {exsec} (x)$	$\sec(x)-1$	-
Excosecant	$\operatorname {excsc} (x)$	$\csc(x)-1$	-
Versine	$\operatorname {versin} (x)$	$1-\cos(x)$	$1-\cos(x)$
Vercosine	$\operatorname {vercosin} (x)$	$1+\cos(x)$	$1+\cos(x)$
Coversine	$\operatorname {coversin} (x)$	$1-\sin(x)$	$1-\sin(x)$
Covercosine	$\operatorname {covercosin} (x)$	$1+\sin(x)$	$1+\sin(x)$
Haversine	$\operatorname {haversin} (x)$	${\frac {1-\cos(x)}{2}}$	${\frac {1}{2}}-{\frac {1}{2}}\cos(x)$
Havercosine	$\operatorname {havercosin} (x)$	${\frac {1+\cos(x)}{2}}$	${\frac {1}{2}}+{\frac {1}{2}}\cos(x)$
Hacoversine	$\operatorname {hacoversin} (x)$	${\frac {1-\sin(x)}{2}}$	${\frac {1}{2}}-{\frac {1}{2}}\sin(x)$
Hacovercosine	$\operatorname {hacovercosin} (x)$	${\frac {1+\sin(x)}{2}}$	${\frac {1}{2}}+{\frac {1}{2}}\sin(x)$
Magnitude of sine wave with amplitude, A, and period, T	-	$A|\sin \left({\frac {2\pi }{T}}x\right)|$	${\frac {4A}{2\pi }}+\sum _{n\,\mathrm {even} }{\frac {-4A}{\pi }}{\frac {1}{1-n^{2}}}\cos({\frac {2\pi n}{T}}x)$ :p. 193

वे फलन जो निष्कोण (चिकने) नहीं हैं

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

नाम	सूत्र	फुर्ये श्रेणी	टीका
त्रिभुज तरंग	${\frac {4}{p}}\left(x-{\frac {p}{2}}\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {2x}{p}}+{\frac {1}{2}}\right\rfloor }$	${\frac {8}{\pi ^{2}}}\sum _{i=0}^{N-1}(-1)^{i}n^{-2}\sin \left({\frac {2\pi nx}{p}}\right)$	non-continuous first derivative
Sawtooth wave	$2\left({\frac {x}{p}}-\left\lfloor {\frac {1}{2}}+{\frac {x}{p}}\right\rfloor \right)$	${\frac {4}{\pi }}\sum _{n\,\mathrm {odd} }^{\infty }{\frac {1}{n}}\sin \left({\frac {n\pi x}{p/2}}\right)$	non-continuous
वर्ग तरंग	$\operatorname {sgn} \left(\sin {\frac {2\pi x}{p}}\right)$	-	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

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