How do I show the Wronskian of $(J_{a}(x),Y_{a}(x)) = \dfrac {2} {\pi x}$
What you're looking for in the first place is a solution to the equation $$ W'(x) = \frac {1}x W(x). $$ Note that you can rewrite this equation as $$ (xW(x))' = W(x) + xW'(x) = 0, $$ which means $$ xW(x) = C, $$ hence $W(x) = c/x$. At this point all you need is to be able to compute the limit of $xW(x) = xW(J_a(X), Y_a(x))$ at only one value $\alpha$ of $\mathbb R$ and/or $\mathbb C$ depending on where you work. If $\alpha = 0$ works fine for you, then so be it ; but another value would work fine too. After you've computed this limit, you know its value is $C$, so you're done.
Hope that helps,
Related videos on Youtube
renagade629
Undergraduate Electrical Engineering and Mathematics student. Aspiring graduate school. I want to understand the theory behind the calculus I am applying in my engineering courses to have a deeper intuition of how they relate.
Updated on August 01, 2022Comments

renagade629 over 1 year
Based of using my undergrad class notes. I know that the wronskian of $(J_{a}(x),Y_{a}(x))$ is $ W(J_{a}(x),Y_{a}(x)) = \left \begin{matrix} J_{a}(x) & Y_{a}(x) \\ J_{a}'(x) & Y_{a}'(x) \end{matrix} \right $ = $J_{a}(x) Y_{a}'(x)J_{a}'(x)Y_{a}(x)$. Where $J_{a}(x)$ is the bessel function of the first kind and is defined as $J_{a}(x) = \dfrac{x} {2}^a \sum_{k=0} ^{\infty} \dfrac{(x^2/4)^k} {k! \Gamma(a+k+1)}$. While $Y_{a}$ is bessels function of the second kind and is defined as $ Y_{a}(x) =\dfrac{J_{a}(x) \cos(\pi a)J_{a}(x)} {\sin(\pi a).}$.I am using the some fact from my class that the wronskian satisfies $\dfrac{dW} {dx}=\dfrac{1} {x} W$ which implies $ W(x) = \dfrac{c} {x}$ for some constant C. (I have no idea where this comes from, I tried to "google" Abel's Identity and found nothing like that. So I am supposed to find $C = lim_{x \rightarrow 0} \; xW(x)$. Then I suppose I divide the limit I get for $c$ to then divide by $x$ to get the wronskian $W(x)$. Am I thinking about this right? I could not figure out the limit of the Wronskian of this series.How should I proceed? I am looking for mathematical manipulations to solve this, no graduate theory will help since I have no background for this. Just a reference sheet of bessel functions and gamma properties.

renagade629 about 11 yearsWhat do you mean by that?

GEdgar about 11 yearsThat's the right answer, but (as we see) you need to provide more information. Or else questioners need to consult a textbook on ODE.