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|}
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))}
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
_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)
