Coshc函數 常見於有關光學散射 [1] 海森堡時空 [2] 雙曲幾何學 的論文 中[3] [4] [5]
Coshc
(
z
)
=
cosh
(
z
)
z
{\displaystyle \operatorname {Coshc} (z)={\frac {\cosh(z)}{z}}}
它是下列微分方程的一個解：
w
(
z
)
z
−
2
d
d
z
w
(
z
)
−
z
d
2
d
z
2
w
(
z
)
=
0
{\displaystyle w\left(z\right)z-2\,{\frac {d}{dz}}w\left(z\right)-z{\frac {d^{2}}{d{z}^{2}}}w\left(z\right)=0}
Coshc 2D plot Coshc'(z) 2D plot 復域虛部
Im
(
cosh
(
x
+
i
y
)
x
+
i
y
)
{\displaystyle \operatorname {Im} \left({\frac {\cosh(x+iy)}{x+iy}}\right)}
復域實部
Re
(
cosh
(
x
+
i
y
)
x
+
i
y
)
{\displaystyle \operatorname {Re} \left({\frac {\cosh \left(x+iy\right)}{x+iy}}\right)}
絕對值
|
cosh
(
x
+
i
y
)
x
+
i
y
|
{\displaystyle \left|{\frac {\cosh(x+iy)}{x+iy}}\right|}
一階導數
sinh
(
z
)
z
−
cosh
(
z
)
z
2
{\displaystyle {\frac {\sinh(z)}{z}}-{\frac {\cosh(z)}{z^{2}}}}
導數實部
−
Re
(
−
1
−
(
cosh
(
x
+
i
y
)
)
2
x
+
i
y
+
cosh
(
x
+
i
y
)
(
x
+
i
y
)
2
)
{\displaystyle -\operatorname {Re} \left(-{\frac {1-(\cosh(x+iy))^{2}}{x+iy}}+{\frac {\cosh(x+iy)}{(x+iy)^{2}}}\right)}
導數虛部
−
Im
(
−
1
−
(
cosh
(
x
+
i
y
)
)
2
x
+
i
y
+
cosh
(
x
+
i
y
)
(
x
+
i
y
)
2
)
{\displaystyle -\operatorname {Im} \left(-{\frac {1-(\cosh(x+iy))^{2}}{x+iy}}+{\frac {\cosh(x+iy)}{(x+iy)^{2}}}\right)}
導數絕對值
|
−
1
−
(
cosh
(
x
+
i
y
)
)
2
x
+
i
y
+
cosh
(
x
+
i
y
)
(
x
+
i
y
)
2
|
{\displaystyle \left|-{\frac {1-(\cosh(x+iy))^{2}}{x+iy}}+{\frac {\cosh(x+iy)}{(x+iy)^{2}}}\right|}
Coshc
(
z
)
=
(
i
z
+
1
/
2
π
)
M
(
1
,
2
,
i
π
−
2
z
)
e
1
/
2
i
π
−
z
z
{\displaystyle \operatorname {Coshc} (z)={\frac {\left(iz+1/2\,\pi \right){{\rm {M}}\left(1,\,2,\,i\pi -2\,z\right)}}{{{\rm {e}}^{1/2\,i\pi -z}}z}}}
Coshc
(
z
)
=
1
2
(
2
i
z
+
π
)
H
e
u
n
B
(
2
,
0
,
0
,
0
,
2
1
/
2
i
π
−
z
)
e
1
/
2
i
π
−
z
z
{\displaystyle \operatorname {Coshc} (z)={\frac {1}{2}}\,{\frac {\left(2\,iz+\pi \right){\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\pi -z}}\right)}{{{\rm {e}}^{1/2\,i\pi -z}}z}}}
Coshc
(
z
)
=
−
i
(
2
i
z
+
π
)
W
h
i
t
t
a
k
e
r
M
(
0
,
1
/
2
,
i
π
−
2
z
)
(
4
i
z
+
2
π
)
z
{\displaystyle \operatorname {Coshc} (z)={\frac {-i\left(2\,iz+\pi \right){{\rm {\mathbf {W} hittakerM}}\left(0,\,1/2,\,i\pi -2\,z\right)}}{\left(4\,iz+2\,\pi \right)z}}}
Coshc
z
≈
(
z
−
1
+
1
2
z
+
1
24
z
3
+
1
720
z
5
+
1
40320
z
7
+
1
3628800
z
9
+
1
479001600
z
11
+
1
87178291200
z
13
+
O
(
z
15
)
)
{\displaystyle \operatorname {Coshc} z\approx ({z}^{-1}+{\frac {1}{2}}z+{\frac {1}{24}}{z}^{3}+{\frac {1}{720}}{z}^{5}+{\frac {1}{40320}}{z}^{7}+{\frac {1}{3628800}}{z}^{9}+{\frac {1}{479001600}}{z}^{11}+{\frac {1}{87178291200}}{z}^{13}+O\left({z}^{15}\right))}
Coshc abs complex 3D Coshc Im complex 3D plot Coshc Re complex 3D plot
Coshc'(z) Im complex 3D plot Coshc'(z) Re complex 3D plot Coshc'(z) abs complex 3D plot
Coshc'(x) abs density plot Coshc'(x) Im density plot Coshc'(x) Re density plot
^ PN Den Outer, TM Nieuwenhuizen, A Lagendijk,Location of objects in multiple-scattering media,JOSA A, Vol. 10, Issue 6, pp. 1209-1218 (1993)
^ T Körpinar ,New characterizations for minimizing energy of biharmonic particles in Heisenberg spacetime - International Journal of Theoretical Physics, 2014 - Springer
^ Nilg¨un S¨onmez,A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry,International Mathematical Forum, 4, 2009, no. 38, 1877 - 1881
^ JHM ten Thije Boonkkamp, J van Dijk, L Liu,Extension of the complete flux scheme to systems of conservation laws,J Sci Comput (2012) 53:552–568,DOI 10.1007/s10915-012-9588-5
^ Weisstein, Eric W. "Coshc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CoshcFunction.html [永久失效連結
_2D_plot.png)
_Im_complex_3D_plot.png)
_Re_complex_3D_plot.png)
_abs_complex_3D_plot.png)
_abs_density_plot.JPG)
_Im_density_plot.JPG)
_Re_density_plot.JPG)