Why is the first type bessel function called first?

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SUMMARY

The first-type Bessel functions are defined as finite at the origin, while the second-type Bessel functions exhibit a singularity at that point. The naming convention arises from the need to identify two independent solutions to the second-order Bessel equation. This classification is not based on any profound mathematical principle but rather on established convention within the field of mathematical analysis.

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what is the difference between first- and second-type bessel functions?
 
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The difference is in the characteristics of their behavior - eg. (and simplistically) the 1st kind are finite at the origin while the 2nd kind have a singularity there for example. So much for the differences - but you also want to know how come the numbering - well, considering these two classes, which one would you like to consider first?

It's a convention. Since the Bessel equation is 2nd order there have to be two independent solutions. I don't think there is anything especially profound about the order.

http://www.springerlink.com/content/rm536p06v8442482/
 

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