The ascending series for J1(z), used by Sir G. B. Airy (_Camb.
Trans._, 1834) in his original investigation of the diffraction of a
circular object-glass, and readily obtained from (6), is

z z^3 z^5 z^7
J1(z) = - - ----- + --------- - ------------- + ... (9).
2 2^2.4 2^2.4^2.6 2^2.4^2.6^2.8

When z is great, we may employ the semi-convergent series
_
/ / 2 \ | 3.5.1 /1\^2
J1(z) = / ( ----- ) sin (z - 1/4[pi]) |1 + ------ ( - )
\/ \[pi]z/ |_ 8.16 \z/
_
3.5.7.9.1.3.5 /1\^4 |
- ------------- ( - ) + ... |
8.16.24.32 \z/ _|
_
/ / 2 \ | 3 1 3.5.7.1.3 /1\ ^3
+ / ( ----- ) cos (z - 1/4[pi]) | - . - - --------- ( - )
\/ \[pi]z/ |_8 z 8.16.24 \z/
_
3.5.7.9.11.1.3.5.7 /1\^5 |
+ ------------------ ( - ) - ... | ... (10).
8.16.24.32.40 \z/ _|

A table of the values of 2z^-1J1(z) has been given by E. C. J. Lommel
(_Schlomilch_, 1870, 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 J1(z) = 0. If these be called z1 z2, z3, ... the radii of the
dark rings in the diffraction pattern are

f[lambda]z1 f[lambda]z2
-----------, -----------, ...
2[pi]R 2[pi]R

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 [phi] so as to obtain the
result for an infinitely thin annular aperture. Thus, if

x = [rho] cos [phi], y = [rho] sin [phi],

_ _ _R _2[pi]
/ / / /
C = | | cos px dx dy = | | cos (p[rho] cos [theta]) [rho]d[rho] d[theta].
_/_/ _/0 _/0

Now by definition

_1/2[pi]
2 / z^2 z^4 z^6
J0(z) = ---- | cos(z cos[theta])d[theta] = --- + ------- - ----------- + ...