Schlömilch function Maple animiation
史咯米尔奇函数(Schlömilch function)是德国数学家史咯米尔奇在1859年首先研究的函数,定义如下:[1]
与其他特殊函数的关系[编辑]
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0287190135393103022d25d8097114928597fbae)
![{\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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/878821490a7cfe5c49b16b837325dae1c0fa8822)
级数展开[编辑]
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bfbedaed69b529d52b63ad7b15071ff2897c8e8)
![{\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})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31a89a2f65af16c78cc05df25ae23312a9c568d0)
参考文献[编辑]
- ^ Schlömilch,Zeitschrift fur Math. und Physik, IV, 1859, p390
- Whittaker and Watson, A Course of Modern Analysis, p352
外部链接[编辑]