Online Encyclopedia

J1 (Z)

Online Encyclopedia
Originally appearing in Volume V08, Page 242 of the 1911 Encyclopedia Britannica.
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J1 (Z) _2 - 22.4+22.42.6 22.42.62.8 + . When z is great, we may employ the semi-convergent series , ~/ r2 1 3 1 _3.5.7.1.3 3.5.7.9.11.1.3.5. I-'V .`zr) co*ā€”170 8 z 8.16.24 (i) + 8.16.24.32.40 7 \i) sā€” ā€¢ (10). A table of the values of 2z-'Jl(z) has been given by E. C. J. Lommel (Schlomilch, 187o, 15, p. 166), to whom is due the first systematic application of Bessel's functions to the diffraction integrals. The illumination vanishes in correspondence with the roots of the equation Jl(z) =o. If these be called z2, z3, ... the radii of the dark rings in the diffraction pattern are 'zi f1z2 2xR ' ark' ' ' being thus inversely proportional to R. The integrations may also be effected by means of polar co-ordinates, taking first the integration with respect to r"so as to obtain the result for an infinitely thin annular aperture. Thus, if x=p cos y=p sin (p, C = ff cos px dx dy = f R f02 r cos (pp cos 0) pdp do. Now by definition
End of Article: J1 (Z)
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