Cauchy-Schlomilch变换
The Cauchy-Schlomilch transformation
摘要：
柯西-施勒米尔希变换指出，对于函数$f$和$a, \, b > 0$，$f(x^{2})$和$af((ax-bx^{-1})^{2}$在区间$[0, \infty)$上的积分相同。这个初等结果被用来计算许多非初等的定积分，其中大多数不能通过符号包获得。应用概率分布也给出。
英文摘要：
The Cauchy-Schlömilch transformation states that for a function $f$ and $a, \, b > 0$, the integral of $f(x^{2})$ and $af((ax-bx^{-1})^{2}$ over the interval $[0, \infty)$ are the same. This elementary result is used to evaluate many non-elementary definite integrals, most of which cannot be obtained by symbolic packages. Applications to probability distributions is also given.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
