Question
Use the recursion relations of Section 15 (for $N$ s as well as for $J '$ 's) and Problem 4 to show that$$J_{n}(x) N_{n+1}(x)-J_{n+1}(x) N_{n}(x)=-\frac{2}{\pi x}$$Hint: Do it first for $n=0$; then use the result in proving the $n=1$ case, and so on.
Step 1
Step 1: We start with the standard results for Bessel's functions and Neumann's functions: $$J'_0(x) = -J_1(x)$$ $$N'_0(x) = -N_1(x)$$ Show more…
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Use the recursion relations of Section 15 (and, as needed, Sections $12,13,17,$ and 20 ) to verify the formulas in Problems 10 to 14. $$\frac{d}{d x} j_{n}(x)=\left[n j_{n-1}(x)-(n+1) j_{n+1}(x)\right] /(2 n+1)$$
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Use the recursion relations of Section 15 (and, as needed, Sections $12,13,17$, and 20 ) to rerifi the formulas in Problems 10 to 14 . $\frac{d}{d x} j_{n}(x)=\left[n j_{n-1}(x)-(n+1) j_{n+1}(x)\right] /(2 n+1)$
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Use the recursion relations of Section 15 (and, as needed, Sections $12,13,17$, and 20 ) to rerifi the formulas in Problems 10 to 14 . Use (15.2) repeatedly to show that $$ J_{1}(x)=x\left(-\frac{1}{x} \frac{d}{d x}\right) J_{0}(x), \quad J_{2}(x)=x^{2}\left(-\frac{1}{x} \frac{d}{d x}\right)^{2} J_{0}(x) $$ and, in general, $$ J_{n}(x)=x^{n}\left(-\frac{1}{x} \frac{d}{d x}\right)^{n} J_{0}(x) $$
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