论文标题
恒星动力学建议的指数积分家族
A family of Exponential Integrals suggested by Stellar Dynamics
论文作者
论文摘要
同时，研究了Chandrasekhar（1943）对具有幂律质量谱和均衡型麦克斯韦·博尔茨曼速度分布的田径恒星的动力摩擦，这是涉及误差函数的一对二维积分，并以指数积分术语为封闭形式（Ciotti Anterals）。在这里我们表明，这两个积分都是（真实）函数$ i（λ，μ，ν; z）家族的非常特殊的案例：= \ int_0^zx^zx^λ\，\ enu（x^μ）\，dx = {γ\左（{1 +λ\\overμ}，z^}，z^firt z^μ\ right） z^{1+λ}\Enu(z^μ)\over 1+λ+ μ(ν-1)}, \quad μ>0,\quad z\geq 0, \eqno (1) $$ where $\Enu$ is the Exponential Integral, $γ$ is the incomplete Euler gamma function, and for existence $λ>\max \ left \ { - 1，-1-μ（ν-1）\ right \} $。只有在一个咨询表中，一个相关的积分才出现，可以将某些工作减少到等式〜（1），而计算机代数系统似乎能够评估封闭（更复杂的）形式的积分，仅提供了某些参数的数值。在这里，我们展示了如何通过基本方法建立等式〜（1）。
While investigating the generalization of the Chandrasekhar (1943) dynamical friction to the case of field stars with a power-law mass spectrum and equipartition Maxwell-Boltzmann velocity distribution, a pair of 2-dimensional integrals involving the Error function occurred, with closed form solution in terms of Exponential Integrals (Ciotti 2010). Here we show that both the integrals are very special cases of the family of (real) functions $$ I(λ,μ,ν; z) :=\int_0^zx^λ\,\Enu(x^μ)\,dx= {γ\left({1+λ\overμ},z^μ\right) + z^{1+λ}\Enu(z^μ)\over 1+λ+ μ(ν-1)}, \quad μ>0,\quad z\geq 0, \eqno (1) $$ where $\Enu$ is the Exponential Integral, $γ$ is the incomplete Euler gamma function, and for existence $λ>\max \left\{-1,-1- μ(ν-1)\right\}$. Only in one of the consulted tables a related integral appears, that with some work can be reduced to eq.~(1), while computer algebra systems seem to be able to evaluate the integral in closed (and more complicated) form only provided numerical values for some of the parameters are assigned. Here we show how eq.~(1) can in fact be established by elementary methods.
