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Tanc function
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In mathematics, the
Tanc function
is defined as
[
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
Imaginary part in complex plane
Im
(
tan
(
x
+
i
y
)
x
+
i
y
)
{\displaystyle \operatorname {Im} \left({\frac {\tan(x+iy)}{x+iy}}\right)}
Real part in complex plane
Re
(
tan
(
x
+
i
y
)
x
+
i
y
)
{\displaystyle \operatorname {Re} \left({\frac {\tan \left(x+iy\right)}{x+iy}}\right)}
absolute magnitude
|
tan
(
x
+
i
y
)
x
+
i
y
|
{\displaystyle \left|{\frac {\tan(x+iy)}{x+iy}}\right|}
First-order derivative
1
−
tan
(
z
)
)
2
z
−
tan
(
z
)
z
2
{\displaystyle {\frac {1-\tan(z))^{2}}{z}}-{\frac {\tan(z)}{z^{2}}}}
Real part of derivative
−
Re
(
−
1
−
(
tan
(
x
+
i
y
)
)
2
x
+
i
y
+
tan
(
x
+
i
y
)
(
x
+
i
y
)
2
)
{\displaystyle -\operatorname {Re} \left(-{\frac {1-(\tan(x+iy))^{2}}{x+iy}}+{\frac {\tan(x+iy)}{(x+iy)^{2}}}\right)}
Imaginary part of derivative
−
Im
(
−
1
−
(
tan
(
x
+
i
y
)
)
2
x
+
i
y
+
tan
(
x
+
i
y
)
(
x
+
i
y
)
2
)
{\displaystyle -\operatorname {Im} \left(-{\frac {1-(\tan(x+iy))^{2}}{x+iy}}+{\frac {\tan(x+iy)}{(x+iy)^{2}}}\right)}
absolute value of derivative
|
−
1
−
(
tan
(
x
+
i
y
)
)
2
x
+
i
y
+
tan
(
x
+
i
y
)
(
x
+
i
y
)
2
|
{\displaystyle \left|-{\frac {1-(\tan(x+iy))^{2}}{x+iy}}+{\frac {\tan(x+iy)}{(x+iy)^{2}}}\right|}
Contents
1
In terms of other special functions
2
Series expansion
3
Gallery
4
See also
5
References
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)}}}}
Tanc
(
z
)
=
2
i
H
e
u
n
B
(
2
,
0
,
0
,
0
,
2
i
z
)
(
2
z
+
π
)
H
e
u
n
B
(
2
,
0
,
0
,
0
,
2
1
/
2
i
(
2
z
+
π
)
)
{\displaystyle \operatorname {Tanc} (z)={\frac {2\,i{\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {iz}}\right)}{\left(2\,z+\pi \right){\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\left(2\,z+\pi \right)}}\right)}}}
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}}\left(0,\,1/2,\,2\,iz\right)}{{{\rm {WhittakerM}}\left(0,\,1/2,\,i\left(2\,z+\pi \right)\right)}z}}}
Series expansion
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 (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\left({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 \left(x\right)}{x}}{dx}=(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\left({z}^{17}\right))}
Gallery
Tanc abs complex 3D
Tanc Im complex 3D plot
Tanc Re complex 3D plot
Tanc'(z) Im complex 3D plot
Tanc'(z) Re complex 3D plot
Tanc'(z) abs complex 3D plot
Tanc abs plot
Tanc Im plot
Tanc Re plot
Tanc'(z) Im plot
Tanc'(z) abs plot
Tanc'(z) Re plot
Tanc integral abs plot
Tanc integral Im plot
Tanc integral Re plot
Tanc abs complex 3D plot
Tanc Im complex 3D plot
Tanc Re complex 3D plot
See also
Sinhc function
Tanhc function
Coshc function
References
^
Weisstein, Eric W. "Tanc Function." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/TancFunction.html
_2D_plot.png)
_Im_complex_3D_plot.png)
_Re_complex_3D_plot.png)
_abs_complex_3D_plot.png)
_Im_plot.JPG)
_abs_plot.JPG)
_Re_plot.JPG)
