史咯米尔奇函数

史咯米尔奇函数（Schlömilch function）是德国数学家史咯米尔奇（英语：Oscar_Schlömilch）在1859年首先研究的函数，定义如下：^[1]

Schlömilch function Maple animiation

${\displaystyle S(v,z)=\int _{0}^{\infty }(1+t)^{-v}e^{-zt}dt=z^{v-1}e^{z}\int _{z}^{\infty }e^{-u}/u^{v}du}$

与其他特殊函数的关系

• ${\displaystyle S(v,z)={\frac {\Gamma (-1+v)*(-1+v)*(-v+z+v*z^{(}-1+v)*exp(z)*\Gamma (-v-1,z)*z^{2}+v^{2}*z^{(}-1+v)*exp(z)*\Gamma (-v-1,z)*z^{2})}{\Gamma (v)*z^{2}}}}$

• ${\displaystyle S(v,z)=({\frac {\Gamma (-1+s)*(-1+s)*(-s+v)}{v^{2}}}+{\frac {(-1+s)*v^{(}-1+s)*exp(v)*\pi }{sin(\pi *s)*(-s+1)}}+{\frac {(-1+s)*(1+s)*s*v^{(}-1+s)*exp(v)*WhittakerW(-1-(1/2)*s,-(1/2)*s-1/2,v)*\Gamma (-1+s)}{v^{(}1+(1/2)*s)*exp((1/2)*v)}}+Pi*v^{(}-1+s)*exp(v)*csc(Pi*s))}$

级数展开

• ${\displaystyle S(0.5,z)\approx {\sqrt {(}}Pi)/{\sqrt {(}}z)-2+{\sqrt {(}}\pi )*{\sqrt {(}}z)-(4/3)*z+(1/2)*z^{(}3/2)*{\sqrt {(}}\pi )-(8/15)*z^{2}+(1/6)*z^{(}5/2)*{\sqrt {(}}\pi )-(16/105)*z^{3}+(1/24)*z^{(}7/2)*{\sqrt {(}}\pi )+O(z^{4})}$

• ${\displaystyle S(0.2,z)\approx {\Gamma (4/5)/z^{(}4/5)+z^{(}1/5)*\Gamma (4/5)+(1/2)*z^{(}6/5)*\Gamma (4/5)+(1/6)*z^{(}11/5)*\Gamma (4/5)+(1/24)*z^{(}16/5)*\Gamma (4/5)-5/4-(25/36)*z-(125/504)*z^{2}-(625/9576)*z^{3}+O(z^{4})}}$

参考文献

1. Schlömilch,Zeitschrift fur Math. und Physik, IV, 1859, p390

• Whittaker and Watson, A Course of Modern Analysis, p352

外部链接

• Schlomilch function （页面存档备份，存于互联网档案馆）