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.
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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 $W$. |
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