Please help me prove the following identity  

$$
a(J\_1(a)Y\_0(a)-J\_0(a)Y\_1(a))=\frac{2}{\pi}
$$
for any $a$.  

$J$ and $Y$ are bessel functions of the first and second kind respectively.
  

Thank you.



Bessel identities
This is equivalent to computing the Wronskian of $J\_0$ and $Y\_0$, since $J'\_0 = -J\_1$ and $Y'\_0 = -Y\_1$.
$$
W(x) =
\begin{vmatrix}
J\_0 & Y\_0 \\\
J'\_0 & Y'\_0
\end{vmatrix} =
\begin{vmatrix}
J\_0 & Y\_0 \\\
-J\_1 & -Y\_1
\end{vmatrix} =
J\_1 Y\_0 - J\_0 Y\_1.
$$
But the zeroth-order Bessel equation is:
$$
\frac{d^2y}{dx^2} + \frac{1}{x}\frac{dy}{dx} + y = 0.
$$
By Abel's formula, $W(a) = \frac{W(1)}{a}$. You now have to compute any value of $W$.


