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In mathematics, the sinhc function appears frequently in papers about optical scattering ,[ 1] and hyperbolic geometry .[ 2] [better source needed ] For
z
≠
0
{\displaystyle z\neq 0}
, it is defined as[ 3] [ 4]
sinhc
(
z
)
=
sinh
(
z
)
z
{\displaystyle \operatorname {sinhc} (z)={\frac {\sinh(z)}{z}}}
The cardinal hyperbolic sine function sinhc(z) plotted in the complex plane from -2-2i to 2+2i
The sinhc function is the hyperbolic analogue of the sinc function , defined by
sin
x
/
x
{\displaystyle \sin x/x}
. 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}
Sinhc 2D plot
Sinhc'(z) 2D plot
Sinhc integral 2D plot
Properties
The first-order derivative is given by
sinhc
′
(
z
)
=
cosh
(
z
)
z
−
sinh
(
z
)
z
2
{\displaystyle \operatorname {sinhc} ^{\prime }(z)={\frac {\cosh(z)}{z}}-{\frac {\sinh(z)}{z^{2}}}}
The Taylor series expansion is
∑
i
=
0
∞
z
2
i
(
2
i
+
1
)
!
.
{\displaystyle \sum _{i=0}^{\infty }{\frac {z^{2i}}{(2i+1)!}}.}
The Padé approximant is
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}}
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}}}}
, where
K
u
m
m
e
r
M
(
a
,
b
,
z
)
{\displaystyle {\rm {KummerM}}(a,b,z)}
is Kummer's confluent hypergeometric function .
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}}}}
, where
H
e
u
n
B
(
q
,
α
,
γ
,
δ
,
ϵ
,
z
)
{\displaystyle {\rm {HeunB}}(q,\alpha ,\gamma ,\delta ,\epsilon ,z)}
is the biconfluent Heun function .
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}}}
, where
W
h
i
t
t
a
k
e
r
M
(
a
,
b
,
z
)
{\displaystyle {\rm {WhittakerM}}(a,b,z)}
is a Whittaker function .
Gallery
Sinhc abs complex 3D
Sinhc Im complex 3D plot
Sinhc Re complex 3D plot
Sinhc'(z) Im complex 3D plot
Sinhc'(z) Re complex 3D plot
Sinhc'(z) abs complex 3D plot
Sinhc abs plot
Sinhc Im plot
Sinhc Re plot
Sinhc'(z) Im plot
Sinhc'(z) abs plot
Sinhc'(z) Re plot
See also
References
^ den Outer, P. N.; Lagendijk, Ad; Nieuwenhuizen, Th. M. (1993-06-01). "Location of objects in multiple-scattering media" . Journal of the Optical Society of America A . 10 (6): 1209. Bibcode :1993JOSAA..10.1209D . doi :10.1364/JOSAA.10.001209 . ISSN 1084-7529 .
^ Nilgün Sönmez, A Trigonometric Proof of the Euler Theorem in Hyperbolic Geometry , International Mathematical Forum, 4, 2009, no. 38, 1877–1881
^ ten Thije Boonkkamp, J. H. M.; van Dijk, J.; Liu, L.; Peerenboom, K. S. C. (2012). "Extension of the Complete Flux Scheme to Systems of Conservation Laws" . Journal of Scientific Computing . 53 (3): 552– 568. doi :10.1007/s10915-012-9588-5 . ISSN 0885-7474 . S2CID 8455136 .
^ Weisstein, Eric W. "Sinhc Function" . mathworld.wolfram.com . Retrieved 2022-11-17 .