查看“︁史咯米尔奇函数”︁的源代码

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[[File:Schlömilch function Maple animation.gif|thumb|300px|Schlömilch function Maple animiation]]
'''史咯米尔奇函数'''（Schlömilch function）是[[德国]][[数学家]]{{le|史咯米尔奇|Oscar_Schlömilch}}在1859年首先研究的函数，定义如下：<ref>Schlömilch,Zeitschrift fur Math. und Physik, IV, 1859, p390</ref>

<math>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</math>

==与其他特殊函数的关系==
*<math> 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   }          </math>

*<math>

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)      )</math>
==级数展开==
*<math>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)</math>

*<math>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)}</math>

==参考文献==
<references/>
*Whittaker and Watson, A Course of Modern Analysis, p352

==外部链接==
*[http://mathworld.wolfram.com/SchloemilchsFunction.html  Schlomilch function] {{Wayback|url=http://mathworld.wolfram.com/SchloemilchsFunction.html |date=20200202130755 }}

[[Category:特殊函数]]