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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:

\begin{displaymath}
\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}
\end{displaymath} (A.1)

From Kreyszig [55] p230:
\begin{displaymath}
J_{-n}(x) = (-1)^{n} J_n(x)
\end{displaymath} (A.2)

and p232:
\begin{displaymath}
\frac{d}{dx}[x^{-\nu}J_{\nu}(x)] = -x^{-\nu}J_{\nu+1}(x)
\end{displaymath} (A.3)


This thesis has moved to Jonathan Kemp Thesis at http://www.kempacoustics.com/thesis
Please change your bookmark/reference to reflect this change as this site may be discontinued
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Next: Projection at a discontinuity Up: thesis Previous: Future work   Contents
Jonathan Kemp 2003-03-24