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## 9.2 Confluent Hypergeometric Functions

### 9.20 Introduction

9.20110 A confluent hypergeometric function is obtained by taking the limit as c→∞ in the solution of Riemann's differential equation

$u=P\left\{\begin{array}{ccc}0& \infty \hfill & c\hfill \\ \frac{1}{2}+\mu & -c\hfill & c-\lambda \hfill & z\hfill \\ \frac{1}{2}-\mu & 0\hfill & \lambda \hfill \end{array}\right\}$ WH

9.202 The equation obtained by means of this limiting process is of the form

1.

$\frac{{\text{d}}^{2}u}{\text{d}{z}^{2}}+\frac{\text{d}u}{\text{d}z}+\left(\frac{\lambda }{z}+\frac{\frac{1}{4}-{\mu }^{2}}{{z}^{2}}\right)u=0$ WH

Equation 9.202 1 has the following two linearly independent solutions:

2.

${z}^{{\frac{1}{2}}^{}+{\mu }_{e}-z}\mathrm{\Phi }\left(\frac{1}{2}+\mu -\lambda ,2\mu +1;z\right)$ 3.

${z}^{}$

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