欧拉反射公式、无穷乘积公式与量子力学的对应原理
Euler's Reflection Formula, Infinite Product Formulas, and the Correspondence Principle of Quantum Mechanics
摘要：
我们推广的沃利斯公式推导$π$从变分计算的光谱的氢原子。我们得到了某些伽玛函数组合的无穷乘积公式，其中包括无理数，如$\sqrt 2$以及一些嵌套的根式。我们还导出了正偶数倒数的欧拉反射公式。我们表明，玻尔的对应原理，使我们能够得到我们的产品公式和反射公式，而不需要的极限定义的伽玛函数。
英文摘要：
We generalize the derivation of the Wallis formula for $π$ from a variational computation of the spectrum of the Hydrogen atom. We obtain infinite product formulas for certain combinations of gamma functions, which include irrational numbers such as $\sqrt 2$ as well as some nested radicals. We also derive Euler's reflection formula for reciprocals of positive even integers. We show that Bohr's correspondence principle allows us to derive our product formulas and the reflection formula without the need for the limit definition of the gamma function.
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。
