सामग्री पर जाएँ

# 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 ${\displaystyle 2\pi }$ समझें। नीचे दिए त्रिकोणमितीय फलनों के लिए,

Un is the nth up/down number,
Bn 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

### वे फलन जो निष्कोण (चिकने) नहीं हैं

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:

## टिप्पणियाँ

1. Formulae are given as Taylor series or derived from other entries.
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.