From Wikipedia, the free encyclopedia
In mathematics, the tanc function is defined for
z
≠
0
{\displaystyle z\neq 0}
[ 1]
tanc
(
z
)
=
tan
(
z
)
z
{\displaystyle \operatorname {tanc} (z)={\frac {\tan(z)}{z}}}
Tanc 2D plot Tanc'(z) 2D plot Tanc integral 2D plot Tanc integral 3D plot
Properties
The first-order derivative of the tanc function is given by
sec
2
(
z
)
z
−
tan
(
z
)
z
2
{\displaystyle {\frac {\sec ^{2}(z)}{z}}-{\frac {\tan(z)}{z^{2}}}}
The Taylor series expansion is
tanc
z
≈
(
1
+
1
3
z
2
+
2
15
z
4
+
17
315
z
6
+
62
2835
z
8
+
1382
155925
z
10
+
21844
6081075
z
12
+
929569
638512875
z
14
+
O
(
z
16
)
)
{\displaystyle \operatorname {tanc} z\approx \left(1+{\frac {1}{3}}z^{2}+{\frac {2}{15}}z^{4}+{\frac {17}{315}}z^{6}+{\frac {62}{2835}}z^{8}+{\frac {1382}{155925}}z^{10}+{\frac {21844}{6081075}}z^{12}+{\frac {929569}{638512875}}z^{14}+O(z^{16})\right)}
∫
0
z
tan
(
x
)
x
d
x
=
(
z
+
1
9
z
3
+
2
75
z
5
+
17
2205
z
7
+
62
25515
z
9
+
1382
1715175
z
11
+
21844
79053975
z
13
+
929569
9577693125
z
15
+
O
(
z
17
)
)
{\displaystyle \int _{0}^{z}{\frac {\tan(x)}{x}}\,dx=\left(z+{\frac {1}{9}}z^{3}+{\frac {2}{75}}z^{5}+{\frac {17}{2205}}z^{7}+{\frac {62}{25515}}z^{9}+{\frac {1382}{1715175}}z^{11}+{\frac {21844}{79053975}}z^{13}+{\frac {929569}{9577693125}}z^{15}+O(z^{17})\right)}
Padé approximant is
tanc
(
z
)
=
(
1
−
7
51
z
2
+
1
255
z
4
−
2
69615
z
6
+
1
34459425
z
8
)
(
1
−
8
17
z
2
+
7
255
z
4
−
4
9945
z
6
+
1
765765
z
8
)
−
1
{\displaystyle \operatorname {tanc} \left(z\right)=\left(1-{\frac {7}{51}}\,{z}^{2}+{\frac {1}{255}}\,{z}^{4}-{\frac {2}{69615}}\,{z}^{6}+{\frac {1}{34459425}}\,{z}^{8}\right)\left(1-{\frac {8}{17}}\,{z}^{2}+{\frac {7}{255}}\,{z}^{4}-{\frac {4}{9945}}\,{z}^{6}+{\frac {1}{765765}}\,{z}^{8}\right)^{-1}}
In terms of other special functions
tanc
(
z
)
=
2
i
K
u
m
m
e
r
M
(
1
,
2
,
2
i
z
)
(
2
z
+
π
)
K
u
m
m
e
r
M
(
1
,
2
,
i
(
2
z
+
π
)
)
{\displaystyle \operatorname {tanc} (z)={\frac {2\,i{{\rm {KummerM}}\left(1,\,2,\,2\,iz\right)}}{\left(2\,z+\pi \right){{\rm {KummerM}}\left(1,\,2,\,i\left(2\,z+\pi \right)\right)}}}}
K
u
m
m
e
r
M
(
a
,
b
,
z
)
{\displaystyle {\rm {KummerM}}(a,b,z)}
confluent hypergeometric function .
tanc
(
z
)
=
2
i
HeunB
(
2
,
0
,
0
,
0
,
2
i
z
)
(
2
z
+
π
)
HeunB
(
2
,
0
,
0
,
0
,
2
(
i
/
2
)
(
2
z
+
π
)
)
{\displaystyle \operatorname {tanc} (z)={\frac {2i\operatorname {HeunB} \left(2,0,0,0,{\sqrt {2}}{\sqrt {iz}}\right)}{(2z+\pi )\operatorname {HeunB} \left(2,0,0,0,{\sqrt {2}}{\sqrt {(i/2)(2z+\pi )}}\right)}}}
H
e
u
n
B
(
q
,
α
,
γ
,
δ
,
ϵ
,
z
)
{\displaystyle {\rm {HeunB}}(q,\alpha ,\gamma ,\delta ,\epsilon ,z)}
Heun function .
tanc
(
z
)
=
W
h
i
t
t
a
k
e
r
M
(
0
,
1
/
2
,
2
i
z
)
W
h
i
t
t
a
k
e
r
M
(
0
,
1
/
2
,
i
(
2
z
+
π
)
)
z
{\displaystyle \operatorname {tanc} (z)={\frac {{\rm {WhittakerM}}(0,\,1/2,\,2\,iz)}{{\rm {WhittakerM}}(0,\,1/2,\,i(2z+\pi ))z}}}
W
h
i
t
t
a
k
e
r
M
(
a
,
b
,
z
)
{\displaystyle {\rm {WhittakerM}}(a,b,z)}
Whittaker function .
Gallery
Tanc abs complex 3D Tanc Im complex 3D plot Tanc Re complex 3D plot
See also
References
^ Weisstein, Eric W. "Tanc Function" . mathworld.wolfram.com . Retrieved 2022-11-17 . 
_2D_plot.png?lang=en)
