where
S = ∫∫ sin(px + qy)dx dy, (3),
C = ∫∫ cos(px + qy)dx dy, (4).
When, as in the application to rectangular or circular apertures, the form is symmetrical with respect to the axes both of x and y, S = 0, and C reduces to
C = ∫∫ cos px cos qy dx dy, (5).
In the case of the circular aperture the distribution of light is of course symmetrical with respect to the focal point p = 0, q = 0; and C is a function of p and q only through √(p² + q²). It is thus sufficient to determine the intensity along the axis of p. Putting q = 0, we get
| C = ∫∫ cos px dx dy = 2 ∫ | +R | cos px √(R² − x²) dx, |
| −R |
R being the radius of the aperture. This integral is the Bessel’s function of order unity, defined by
| J1(z) = | z | ∫ | π | cos (z cos φ) sin² φ dφ (6). |
| π | 0 |
Thus, if x = R cos φ,