Name
Symbol
Formula [ nb 1]
Fourier Series
Sine
sin
(
x
)
{\displaystyle \sin(x)}
∑
n
=
0
∞
(
−
1
)
n
x
2
n
+
1
(
2
n
+
1
)
!
{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}}
sin
(
x
)
{\displaystyle \sin(x)}
cas (mathematics)
cas
(
x
)
{\displaystyle \operatorname {cas} (x)}
sin
(
x
)
+
cos
(
x
)
{\displaystyle \sin(x)+\cos(x)}
sin
(
x
)
+
cos
(
x
)
{\displaystyle \sin(x)+\cos(x)}
Cosine
cos
(
x
)
{\displaystyle \cos(x)}
∑
n
=
0
∞
(
−
1
)
n
x
2
n
(
2
n
)
!
{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}}
cos
(
x
)
{\displaystyle \cos(x)}
cis (mathematics)
e
i
x
,
cis
(
x
)
{\displaystyle e^{ix},\operatorname {cis} (x)}
cos(x ) + i sin(x )
cos
(
x
)
+
i
sin
(
x
)
{\displaystyle \cos(x)+i\sin(x)}
Tangent
tan
(
x
)
{\displaystyle \tan(x)}
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)!}}}
2
∑
n
=
1
∞
(
−
1
)
n
−
1
sin
(
2
n
x
)
{\displaystyle 2\sum _{n=1}^{\infty }(-1)^{n-1}\sin(2nx)}
[ 1]
Cotangent
cot
(
x
)
{\displaystyle \cot(x)}
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)!}}}
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)}
[citation needed ]
Secant
sec
(
x
)
{\displaystyle \sec(x)}
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)!}}}
-
Cosecant
csc
(
x
)
{\displaystyle \csc(x)}
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)!}}}
-
Exsecant
exsec
(
x
)
{\displaystyle \operatorname {exsec} (x)}
sec
(
x
)
−
1
{\displaystyle \sec(x)-1}
-
Excosecant
excsc
(
x
)
{\displaystyle \operatorname {excsc} (x)}
csc
(
x
)
−
1
{\displaystyle \csc(x)-1}
-
Versine
versin
(
x
)
{\displaystyle \operatorname {versin} (x)}
1
−
cos
(
x
)
{\displaystyle 1-\cos(x)}
1
−
cos
(
x
)
{\displaystyle 1-\cos(x)}
Vercosine
vercosin
(
x
)
{\displaystyle \operatorname {vercosin} (x)}
1
+
cos
(
x
)
{\displaystyle 1+\cos(x)}
1
+
cos
(
x
)
{\displaystyle 1+\cos(x)}
Coversine
coversin
(
x
)
{\displaystyle \operatorname {coversin} (x)}
1
−
sin
(
x
)
{\displaystyle 1-\sin(x)}
1
−
sin
(
x
)
{\displaystyle 1-\sin(x)}
Covercosine
covercosin
(
x
)
{\displaystyle \operatorname {covercosin} (x)}
1
+
sin
(
x
)
{\displaystyle 1+\sin(x)}
1
+
sin
(
x
)
{\displaystyle 1+\sin(x)}
Haversine
haversin
(
x
)
{\displaystyle \operatorname {haversin} (x)}
1
−
cos
(
x
)
2
{\displaystyle {\frac {1-\cos(x)}{2}}}
1
2
−
1
2
cos
(
x
)
{\displaystyle {\frac {1}{2}}-{\frac {1}{2}}\cos(x)}
Havercosine
havercosin
(
x
)
{\displaystyle \operatorname {havercosin} (x)}
1
+
cos
(
x
)
2
{\displaystyle {\frac {1+\cos(x)}{2}}}
1
2
+
1
2
cos
(
x
)
{\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\cos(x)}
Hacoversine
hacoversin
(
x
)
{\displaystyle \operatorname {hacoversin} (x)}
1
−
sin
(
x
)
2
{\displaystyle {\frac {1-\sin(x)}{2}}}
1
2
−
1
2
sin
(
x
)
{\displaystyle {\frac {1}{2}}-{\frac {1}{2}}\sin(x)}
Hacovercosine
hacovercosin
(
x
)
{\displaystyle \operatorname {hacovercosin} (x)}
1
+
sin
(
x
)
2
{\displaystyle {\frac {1+\sin(x)}{2}}}
1
2
+
1
2
sin
(
x
)
{\displaystyle {\frac {1}{2}}+{\frac {1}{2}}\sin(x)}
Jacobi elliptic function sn
sn
(
x
,
m
)
{\displaystyle \operatorname {sn} (x,m)}
sin
am
(
x
,
m
)
{\displaystyle \sin \operatorname {am} (x,m)}
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)}}}
Jacobi elliptic function cn
cn
(
x
,
m
)
{\displaystyle \operatorname {cn} (x,m)}
cos
am
(
x
,
m
)
{\displaystyle \cos \operatorname {am} (x,m)}
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)}}}
Jacobi elliptic function dn
dn
(
x
,
m
)
{\displaystyle \operatorname {dn} (x,m)}
1
−
m
sn
2
(
x
,
m
)
{\displaystyle {\sqrt {1-m\operatorname {sn} ^{2}(x,m)}}}
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)}}}
Jacobi elliptic function zn
zn
(
x
,
m
)
{\displaystyle \operatorname {zn} (x,m)}
∫
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}
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)}}}
Weierstrass elliptic function
℘
(
x
,
Λ
)
{\displaystyle \wp (x,\Lambda )}
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]}
{\displaystyle }
Clausen function
Cl
2
(
x
)
{\displaystyle \operatorname {Cl} _{2}(x)}
−
∫
0
x
ln
|
2
sin
t
2
|
d
t
{\displaystyle -\int _{0}^{x}\ln \left|2\sin {\frac {t}{2}}\right|dt}
∑
k
=
1
∞
sin
k
x
k
2
{\displaystyle \sum _{k=1}^{\infty }{\frac {\sin kx}{k^{2}}}}