The second solution to Bessel’s equation is \(Y_m(x)\), the Bessel function of the second kind. Compute the function \(Y_m(x)\) using upward recursion $$ Y_{m+1}=\frac{2m}{x}Y_m(x)-Y_{m-1}(x). $$ To obtain the starting values for recursion, use the identities $$ Y_0(x) = \frac{2}{\pi} \left[ \ln{\frac{x}{2}} + \gamma \right] J_0(x) - \frac{4}{\pi} \sum^\infty_{k=1}(-1)^k \frac{J_{2k}(x)}{k} $$ and $$ J_1(x)Y_0(x)-J_0(x)Y_1(x) = \frac{2}{\pi x} $$ where \(\gamma \approx 0.577215664\). Demonstrate your routine by producing plots of \(Y_m(x)\) for \(0 < x < 50\) and various \(m\).
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Numerical Integration Result
