Sinhc函数

数学上的Sinhc函数定义如下^[1][2][3]

${\displaystyle \operatorname {Sinhc} (z)={\frac {\sinh(z)}{z}}}$，它是下列微分方程的一个解：

${\displaystyle w(z)z-2\,{\frac {d}{dz}}w(z)-z{\frac {d^{2}}{dz^{2}}}w(z)=0}$

Sinhc 2D plot

Sinhc'(z) 2D plot

Sinhc integral 2D plot

复域虚部

• ${\displaystyle \operatorname {Im} \left({\frac {\sinh(x+iy)}{x+iy}}\right)}$

复域实部

• ${\displaystyle \operatorname {Re} \left({\frac {\sinh \left(x+iy\right)}{x+iy}}\right)}$

绝对值

• ${\displaystyle \left|{\frac {\sinh(x+iy)}{x+iy}}\right|}$

一阶导数

• ${\displaystyle {\frac {\cosh(z)}{z}}-{\frac {\sinh(z)}{z^{2}}}}$

导数实部

• ${\displaystyle -\operatorname {Re} \left(-{\frac {1-(\sinh(x+iy))^{2}}{x+iy}}+{\frac {\sinh(x+iy)}{(x+iy)^{2}}}\right)}$

导数虚部

• ${\displaystyle -\operatorname {Im} \left(-{\frac {1-(\sinh(x+iy))^{2}}{x+iy}}+{\frac {\sinh(x+iy)}{(x+iy)^{2}}}\right)}$

绝对值

• ${\displaystyle \left|-{\frac {1-(\sinh(x+iy))^{2}}{x+iy}}+{\frac {\sinh(x+iy)}{(x+iy)^{2}}}\right|}$

表示为其他特殊函数

• ${\displaystyle \operatorname {Sinhc} (z)={\frac {{\rm {KummerM}}\left(1,\,2,\,2\,z\right)}{{\rm {e}}^{z}}}}$

• ${\displaystyle \operatorname {Sinhc} (z)={\frac {{\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {z}}\right)}{{\rm {e}}^{z}}}}$

• ${\displaystyle \operatorname {Sinhc} (z)=1/2\,{\frac {{\rm {WhittakerM}}\left(0,\,1/2,\,2\,z\right)}{z}}}$

级数展开

${\displaystyle \operatorname {Sinhc} z\approx (1+{\frac {1}{3}}{z}^{2}+{\frac {2}{15}}{z}^{4}+{\frac {17}{315}}{z}^{6}+{\frac {62}{2835}}{z}^{8}+{\frac {1382}{155925}}{z}^{10}+{\frac {21844}{6081075}}{z}^{12}+{\frac {929569}{638512875}}{z}^{14}+O\left({z}^{16}\right))}$

图集

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

参见

• Tanhc函数
• Coshc函数
• Tanc函数
• 双曲正弦积分函数

参考文献

1. 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
2. Weisstein, Eric W. "Sinhc Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SinhcFunction.html （页面存档备份，存于互联网档案馆）
3. P. N. den Outer, Th. M. Nieuwenhuizen, and Ad Lagendijk,Location of objects in multiple-scattering media,JOSA A, Vol. 10, Issue 6, pp. 1209-1218 (1993)