Abstract

The ath power of x, denoted by x^a, might be defined as “the number obtained by multiplying unity a times with x”. Thus (2.5)³ = 2.5 x 2.5 x 2.5. This definition makes sense only when a is an integer. However, it is an elementary fact that the definition of the power of a number can be meaningfully and consistently extended to real anil even complex values of a. Likewise, the original definition of the derivative of a function makes sense only for integral orders, i.e. we can speak of the first or second derivative and so on. However, it is possible to extend the definition of the derivative to noninteger orders by using an elementary properly of Fourier transforms. Bracewell shows how fractional derivatives can be used to characterize the discontinuities of certain funct ions [1]. An example from the field of optics is related to the Talbot effect [2], in which self-images of an input object are observed at the 2-D planes z — N_z0 for integer N (z is the axial coordinate and z₀ a characteristic distance). Using a self-transformation technique [3], it was shown that N could also take on certain rational values

© 1993 Optical Society of America