My Quantum Mechanics textbook (John S. Townsend's A Modern Approach to Quantum Mechanics) mentions regular solutions and irregular solutions. It claims that regular solutions (at the origin) to the spherical Bessel equation are called spherical Bessel functions, while its irregular solutions (at the origin) are called spherical Neumann functions.

What's the difference between "regular" and "irregular"? And, if possible, please also tell me what "regular" and "irregular" solutions are in general (i.e.: please don't just tell me how they differ without illuminating what they are in the first place).

See below for a photo of the relevant page in my textbook. The stuff I'm asking about is found below eqn. 10.67.enter image description here


2 Answers 2


Regular functions are well defined (finite). Irregular functions tend to infinity in the limit of approaching some point.

In this case, all the Bessel functions tend to zero (except j0 which goes to 1) as you approach the origin. The Neumann functions approach +infinity as you approach the origin from the positive side.

  • $\begingroup$ That's it? Why would one use what seems to me unnecessarily obscure language for this? $\endgroup$ Apr 11, 2016 at 1:48
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    $\begingroup$ What's counts as unnecessarily obscure is in the eyes of the beholder. $\endgroup$
    – garyp
    Apr 11, 2016 at 1:59
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    $\begingroup$ "what seems to me" :P $\endgroup$ Apr 11, 2016 at 2:07

I think that a regular solution $y(k,r)$ is one that satisfies the boundary conditions, $$y(k,0) = 0$$ $$y'(k,0) = 1,$$ while a irregular solution does not. However, there may be more to the answer than this.


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