In mathematics, the Sinhc function appears frequently in papers about optical scattering,[1] Heisenberg Spacetime[2] and hyperbolic geometry.[3] It is defined as[4] [5]
Sinhc
(
z
)
=
sinh
(
z
)
z
{\displaystyle \operatorname {Sinhc} (z)={\frac {\sinh(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
(
sinh
(
x
+
i
y
)
x
+
i
y
)
{\displaystyle \operatorname {Im} \left({\frac {\sinh(x+iy)}{x+iy}}\right)}
Real part in complex plane
Re
(
sinh
(
x
+
i
y
)
x
+
i
y
)
{\displaystyle \operatorname {Re} \left({\frac {\sinh(x+iy)}{x+iy}}\right)}
absolute magnitude
|
sinh
(
x
+
i
y
)
x
+
i
y
|
{\displaystyle \left|{\frac {\sinh(x+iy)}{x+iy}}\right|}
First-order derivative
cosh
(
z
)
z
−
sinh
(
z
)
z
2
{\displaystyle {\frac {\cosh(z)}{z}}-{\frac {\sinh(z)}{z^{2}}}}
Real part of derivative
−
Re
(
−
1
−
(
sinh
(
x
+
i
y
)
)
2
x
+
i
y
+
sinh
(
x
+
i
y
)
(
x
+
i
y
)
2
)
{\displaystyle -\operatorname {Re} \left(-{\frac {1-(\sinh(x+iy))^{2}}{x+iy}}+{\frac {\sinh(x+iy)}{(x+iy)^{2}}}\right)}
Imaginary part of derivative
−
Im
(
−
1
−
(
sinh
(
x
+
i
y
)
)
2
x
+
i
y
+
sinh
(
x
+
i
y
)
(
x
+
i
y
)
2
)
{\displaystyle -\operatorname {Im} \left(-{\frac {1-(\sinh(x+iy))^{2}}{x+iy}}+{\frac {\sinh(x+iy)}{(x+iy)^{2}}}\right)}
absolute value of derivative
|
−
1
−
(
sinh
(
x
+
i
y
)
)
2
x
+
i
y
+
sinh
(
x
+
i
y
)
(
x
+
i
y
)
2
|
{\displaystyle \left|-{\frac {1-(\sinh(x+iy))^{2}}{x+iy}}+{\frac {\sinh(x+iy)}{(x+iy)^{2}}}\right|}
In terms of other special functions [ ]
Sinhc
(
z
)
=
K
u
m
m
e
r
M
(
1
,
2
,
2
z
)
e
z
{\displaystyle \operatorname {Sinhc} (z)={\frac {{\rm {KummerM}}(1,\,2,\,2\,z)}{e^{z}}}}
Sinhc
(
z
)
=
HeunB
(
2
,
0
,
0
,
0
,
2
z
)
e
z
{\displaystyle \operatorname {Sinhc} (z)={\frac {\operatorname {HeunB} \left(2,0,0,0,{\sqrt {2}}{\sqrt {z}}\right)}{e^{z}}}}
Sinhc
(
z
)
=
1
/
2
W
h
i
t
t
a
k
e
r
M
(
0
,
1
/
2
,
2
z
)
z
{\displaystyle \operatorname {Sinhc} (z)=1/2\,{\frac {{\rm {WhittakerM}}(0,\,1/2,\,2\,z)}{z}}}
Series expansion [ ]
∑
i
=
0
∞
z
2
i
(
2
i
+
1
)
!
.
{\displaystyle \sum _{i=0}^{\infty }{\frac {z^{2i}}{(2i+1)!}}.}
Padé approximation [ ]
Sinhc
(
z
)
=
(
1
+
53272705
360869676
z
2
+
38518909
7217393520
z
4
+
269197963
3940696861920
z
6
+
4585922449
15605159573203200
z
8
)
(
1
−
2290747
120289892
z
2
+
1281433
7217393520
z
4
−
560401
562956694560
z
6
+
1029037
346781323848960
z
8
)
−
1
{\displaystyle \operatorname {Sinhc} \left(z\right)=\left(1+{\frac {53272705}{360869676}}\,{z}^{2}+{\frac {38518909}{7217393520}}\,{z}^{4}+{\frac {269197963}{3940696861920}}\,{z}^{6}+{\frac {4585922449}{15605159573203200}}\,{z}^{8}\right)\left(1-{\frac {2290747}{120289892}}\,{z}^{2}+{\frac {1281433}{7217393520}}\,{z}^{4}-{\frac {560401}{562956694560}}\,{z}^{6}+{\frac {1029037}{346781323848960}}\,{z}^{8}\right)^{-1}}
Gallery [ ]
Sinhc'(z) Im complex 3D plot
Sinhc'(z) Re complex 3D plot
Sinhc'(z) abs complex 3D 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
^ 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. "Sinhc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SinhcFunction.html