Properties of Bessel functions

Table A.1: Zeros of the Bessel function $J_1$ to 7 decimal places

n	$\gamma_n$	n	$\gamma_n$	n	$\gamma_n$
0	0.0000000	16	51.0435352	32	101.3126618
1	3.8317060	17	54.1855536	33	104.4543658
2	7.0155867	18	57.3275254	34	107.5960633
3	10.1734681	19	60.4694578	35	110.7377548
4	13.3236919	20	63.6113567	36	113.8794408
5	16.4706301	21	66.7532267	37	117.0211219
6	19.6158585	22	69.8950718	38	120.1627983
7	22.7600844	23	73.0368952	39	123.3044705
8	25.9036721	24	76.1786996	40	126.4460139
9	29.0468285	25	79.3204872	41	129.5878033
10	32.1896799	26	82.4622599	42	132.7294644
11	35.3323076	27	85.6040194	43	135.8711224
12	38.4747662	28	88.7457671	44	139.0127774
13	41.6170942	29	91.8875043	45	142.1544297
14	44.7593190	30	95.0292318
15	47.9014609	31	98.1709507

Standard integral from Jahnke and Emde [73] p146:

$$\int x J_p(\alpha x) J_p(\beta x) dx = \frac{\beta x J_p(\... ...alpha x J_{p-1}(\alpha x) J_p(\beta x)} {\alpha^2 - \beta^2}$$ (A.1)

From Kreyszig [55] p230:

$$J_{-n}(x) = (-1)^{n} J_n(x)$$ (A.2)

and p232:

$$\frac{d}{dx}[x^{-\nu}J_{\nu}(x)] = -x^{-\nu}J_{\nu+1}(x)$$ (A.3)