库默尔方程和合流超几何函数解的物理学家指南
A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions
摘要：
合流超几何方程，又称库默尔方程，是物理学、化学和工程学中最重要的微分方程之一。它的两个幂级数解是库默尔函数M（a，b，z），通常被称为第一类合流超几何函数，和z^{1-b}M（1+a-b，2-b，z），其中a和b是微分方程中出现的参数。第三个函数，Tricomi函数，U（a，b，z），有时被称为第二类合流超几何函数，也是通常使用的合流超几何方程的解。所有这三个功能必须考虑在寻找两个线性无关的解决方案的合流超几何方程。当a、b和a-b是整数时，存在这样的情况，其中这些函数中的一个没有被定义，或者其中两个函数不是线性无关的，或者微分方程的线性无关解之一不同于这三个函数。这些特殊情况中的许多情况正好对应于解决物理问题所需的情况。这导致了关于如何处理合流超几何方程的严重混乱，尽管有权威的参考文献，如NIST数学函数数字图书馆。在这里，我们仔细地描述了所有不同的情况下，必须考虑和显式公式是什么两个线性无关的解决方案的合流超几何方程。我们的结果总结在第3节的表I中。作为一个例子，我们使用这些解决方案来研究束缚态的氢原子，超出了标准的处理在教科书中。我们还简要地考虑了截止库仑势。我们希望本指南将有助于物理教学，涉及合流超几何微分方程。
英文摘要：
The confluent hypergeometric equation, also known as Kummer's equation, is one of the most important differential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z), often referred to as the confluent hypergeometric function of the first kind, and z^{1-b}M(1+a-b,2-b,z), where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, U(a,b,z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. All three of these functions must be considered in a search for two linearly independent solutions of the confluent hypergeometric equation. There are situations, when a, b, and a - b are integers, where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independent solutions of the differential equation is different from these three functions. Many of these special cases correspond precisely to cases needed to solve physics problems. This leads to significant confusion about how to work with confluent hypergeometric equations, in spite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. Our results are summarized in Table I in Section 3. As an example, we use these solutions to study the bound states of the hydrogenic atom, going beyond the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physics instruction that involves the confluent hypergeometric differential equation.
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