In mathematics, the Coshc function appears frequently in papers about optical scattering,[1] [2] [3] [4] [5]
Coshc
(
z
)
=
cosh
(
z
)
z
{\displaystyle \operatorname {Coshc} (z)={\frac {\cosh(z)}{z}}}
It is a solution of the following differential equation:
w
(
z
)
z
−
2
d
d
z
w
(
z
)
−
z
d
2
d
z
2
w
(
z
)
=
0
{\displaystyle w(z)z-2{\frac {d}{dz}}w(z)-z{\frac {d^{2}}{dz^{2}}}w(z)=0}
Imaginary part in complex plane
Im
(
cosh
(
x
+
i
y
)
x
+
i
y
)
{\displaystyle \operatorname {Im} \left({\frac {\cosh(x+iy)}{x+iy}}\right)}
Real part in complex plane
Re
(
cosh
(
x
+
i
y
)
x
+
i
y
)
{\displaystyle \operatorname {Re} \left({\frac {\cosh(x+iy)}{x+iy}}\right)}
absolute magnitude
|
cosh
(
x
+
i
y
)
x
+
i
y
|
{\displaystyle \left|{\frac {\cosh(x+iy)}{x+iy}}\right|}
First-order derivative
sinh
(
z
)
z
−
cosh
(
z
)
z
2
{\displaystyle {\frac {\sinh(z)}{z}}-{\frac {\cosh(z)}{z^{2}}}}
Real part of derivative
−
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)}
Imaginary part of derivative
−
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)}
absolute value of derivative
|
−
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|}
In terms of other special functions [ ]
Coshc
(
z
)
=
(
i
z
+
1
/
2
π
)
M
(
1
,
2
,
i
π
−
2
z
)
e
(
i
/
2
)
π
−
z
z
{\displaystyle \operatorname {Coshc} (z)={\frac {(iz+1/2\,\pi ){\rm {M}}(1,2,i\pi -2z)}{e^{(i/2)\pi -z}z}}}
Coshc
(
z
)
=
1
2
(
2
i
z
+
π
)
HeunB
(
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 {(2\,iz+\pi )\operatorname {HeunB} \left(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\pi -z}}\right)}{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(2\,iz+\pi ){{\rm {\mathbf {W} hittakerM}}(0,\,1/2,\,i\pi -2z)}}{(4iz+2\pi )z}}}
Series expansion [ ]
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 \left(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(z^{15})\right)}
Padé approximation [ ]
Coshc
(
z
)
=
23594700729600
+
11275015752000
z
2
+
727718024880
z
4
+
13853547000
z
6
+
80737373
z
8
147173
z
9
−
39328920
z
7
+
5772800880
z
5
−
522334612800
z
3
+
23594700729600
z
{\displaystyle \operatorname {Coshc} \left(z\right)={\frac {23594700729600+11275015752000\,{z}^{2}+727718024880\,{z}^{4}+13853547000\,{z}^{6}+80737373\,{z}^{8}}{147173\,{z}^{9}-39328920\,{z}^{7}+5772800880\,{z}^{5}-522334612800\,{z}^{3}+23594700729600\,z}}}
Gallery [ ]
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
See also [ ] References [ ]
^ 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
^ Nilgün Sönmez, 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 [permanent dead link
