Sinhc函数

数学上的Sinhc函数定义如下^[1]^[2]^[3]

\operatorname {Sinhc} (z)={\frac {\sinh(z)}{z}}

，它是下列微分方程的一个解：

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

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

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

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

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

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

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

\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))

[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)

表示为其他特殊函数