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This is clearly incorrect. The correct series is
This is clearly incorrect. The correct series is
<math>\sum_{i=0}^\infty \frac{z^{2i}}{(2i+1)!}</math>.
: <math>\sum_{i=0}^\infty \frac{z^{2i}}{(2i+1)!}</math>.
==Padé approximation==
==Padé approximation==
Revision as of 21:51, 22 April 2021
In mathematics, the Sinhc function appears frequently in papers about optical scattering,[1] [2] [3] [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}
Sinhc 2D plot Sinhc'(z) 2D plot Sinhc integral 2D plot 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)}{{\rm {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)}{{\rm {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
Sinhc
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 {Sinhc} z\approx \left(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(z^{16})\right)}
This is clearly incorrect. The correct series is
∑
i
=
0
∞
z
2
i
(
2
i
+
1
)
!
{\displaystyle \sum _{i=0}^{\infty }{\frac {z^{2i}}{(2i+1)!}}}
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 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
^ 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
_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)