| ƒλz1 | ƒλz2 | , ... | |
| 2πR | 2πR |
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 φ so as to obtain the result for an infinitely thin annular aperture. Thus, if
x = ρ cos φ, y = ρ sin φ,
| C = ∫∫ cos px dx dy = ∫ | R | ∫ | 2π | cos (pρ cos θ) ρdρ dθ. |
| 0 | 0 |
Now by definition
| J0(z) = | 2 | ∫ | ½π | cos (z cos θ) dθ = 1 − | z² | + | z4 | − | z6 | + ... (11). |
| π | 0 | 2² | 2²·4² | 2²·4²·6² |
The value of C for an annular aperture of radius r and width dr is thus
dC = 2 π J0 (pρ) ρ dρ, (12).
For the complete circle,