Origem: Wikipédia, a enciclopédia livre.
Em matemática , a função Sinhc aparece frequentemente em artigos sobre dispersão óptica , [ 1] [ 2] geometria hiperbólica . [ 3] [ 4] [ 5]
Sinhc
(
z
)
=
sinh
(
z
)
z
{\displaystyle \operatorname {Sinhc} (z)={\frac {\sinh(z)}{z}}}
É uma solução da seguinte equação diferencial :
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}
Gráfico 2D de Sinhc Gráfico 2D de Sinhc'(z) Gráfico 2D da Integral de Sinhc Parte imaginária no plano complexo
Im
(
sinh
(
x
+
i
y
)
x
+
i
y
)
{\displaystyle \operatorname {Im} \left({\frac {\sinh(x+iy)}{x+iy}}\right)}
Parte real no plano complexo
Re
(
sinh
(
x
+
i
y
)
x
+
i
y
)
{\displaystyle \operatorname {Re} \left({\frac {\sinh(x+iy)}{x+iy}}\right)}
Magnitude absoluta
|
sinh
(
x
+
i
y
)
x
+
i
y
|
{\displaystyle \left|{\frac {\sinh(x+iy)}{x+iy}}\right|}
Derivada de primeira ordem
cosh
(
z
)
z
−
sinh
(
z
)
z
2
{\displaystyle {\frac {\cosh(z)}{z}}-{\frac {\sinh(z)}{z^{2}}}}
Parte real da derivada
−
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)}
Parte imaginária da derivada
−
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)}
Valor absoluto da derivada
|
−
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|}
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}}}
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)}
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}}
Gráfico 3D de abs(Sinhc) Gráfico 3D de Im(Sinhc) Gráfico 3D de Re(Sinhc)
Gráfico 3D de Im(Sinhc'(z)) Gráfico 3D de Re(Sinhc'(z)) Gráfico 3D de abs(Sinhc'(z))
Gráfico 2D de abs(Sinhc) Gráfico 2D de Im(Sinhc) Gráfico 2D de Re(Sinhc)
Gráfico 2D de Im(Sinhc'(z)) Gráfico 2D de abs(Sinhc'(z)) Gráfico 2D de Re(Sinhc '(z))
↑ 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)