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I'm working through "Elementary Differential Equations", 10th Ed (Boyce/DiPrima) on my own -- 120 miles from the nearest easily accessible prof or grad student -- and I've a question on a technique in the text for getting the equation for Bessel's Equations of the second kind (Section 5.7, pp. 298).

The previous section laid out a mechanism for determining the second solution where roots were equal or differed by an integer. When equal (r1=r2), the second solution includes an infinite series representation with a different set of coefficients (call them [bn], as opposed to the first solution coefficients, [an]). That theorem (5.6.1) gave no specifics about finding those coefficients, except to use "... the recurrence relationship" (similarly, I assume, to finding them in the first solution).

In this next section, on Bessel's Equation, they introduce the technique (for Order Zero equations) of using the ratio between the first derivative of the function from the first equation relating the root to the coefficients -- call it an(r) -- and the equation itself: that is a'n(r)/an(r). And using that relationship, just utilize a'n(r) as the bn coefficients. But I can nowhere in the text find the justification for this technique. I looked -- I believe carefully -- through all the previous sections, and I've searched the web moderately thoroughly (looking at published resources from several university math programs), and I cannot find this technique referenced or explained.

Can anyone point me to where I might look to understand why this is a useful approach to finding the coefficients in the Bessel Equation of the Second Kind (of Order Zero)? And if it is easily explained, I'd certainly benefit from your justification (though I understand that might be time consuming, and would be happy with just a reference!).

Thanks!

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There is an excellent book called A Treatise on the Theory of Bessel Functions by G. N. Watson.

The first chapter is the history of Bessel functions, mostly about how Bessel himself was studying the differential equations which lead to the solution. It then goes on to find the coefficients for the infinite series.

The third chapter is simply called "Bessel Functions" and it talks about how others have extended the solution found by Bessel, most importantly Hankel, who discovered the second solution that you are after.

I'll not put too much detail, but the argument is along the following lines.

Bessels differential equation is, $$ z^2 \frac{d^2y}{dz^2} + z \frac{dy}{dz} + (z^2 - \nu^2) y = 0 $$ of which the Bessel function $y(z) = J_\nu(z)$ is a solution. Hankel was looking for solutions related to, $$ J_\nu(z) - (-1)^nJ_{-\nu}(z) $$ and found that, $$ Y_n(z) = \lim_{\nu \to n}\frac{J_\nu(z) - (-)^n J_\nu(z)}{\nu - n} $$ is a solution as well. Taking a little bit of care with differentiating the complex analytic function we can arrive at, $$ \begin{align} Y_{-n}(z) =& (-1)^nY_n(z) \\ Y_0(z) =& 2 \left[ \frac{\partial J_\nu(z)}{\partial \nu} \right]_{\nu = 0} \end{align} $$ At this point we can substitute in the infinite series for the first Bessel function, which gives, $$ \begin{align} Y_0(z) =& 2 \left[\frac{\partial}{\partial \nu} \sum_{m=0}^\infty \frac{(-1)^m (\frac{1}{2} z)^{\nu +2m}}{m!\Gamma(\nu + m + 1)} \right] \\ =& 2 \sum_{m=0}^\infty \frac{(-1)^m(\frac{1}{2}z)^{\nu + 2m}}{(m!)^2} \{ \log\frac{1}{2}z - \psi(m+1) \} \end{align} $$ where $\psi$ denotes the digamma function.

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  • $\begingroup$ Thank you SO much! Wonderful, enlightening response. And I shall surely get that book! All my best to you. $\endgroup$ – Larry Cosner Mar 4 '18 at 14:54

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