5-2] FOURIER ANALYSIS 239 



We are familiar with the representation of periodic functions by Fourier 

 series. A Fourier integral is a limiting case of such a series where the period 

 becomes indefinitely long. The separation between components becomes 

 indefinitely small as do their magnitudes. For properly restricted functions, 

 however, the magnitude density possesses well-defined values and a Fourier 

 integral exists. The restrictions on a function /(/) in order that it have a 

 Fourier integral are that the integrals of both its square and its absolute 

 value have finite values and that it possess only a finite number of dis- 

 continuities in any finite interval. When these conditions are met, a 

 function F{co) can be defined by the relation 



/. 



F{c^) = / Me-'-'^'dL (5-1) 



If we suppose that/(/) is a function of time, then F{oci) is the spectrum of 

 /(/) and gives the density of its diflPerential frequency components in much 

 the same way that a Fourier series gives the resolution of a periodic function 

 into finite frequency components. The variable co is the angular frequency 

 equal to 27r times the cyclical frequency. In general, F(ci;) may be complex. 



The time function /(/) is given by the integral of all the differential 

 Fourier components in a manner very similar to the way in which the sum 

 of all the components of a Fourier series represents a periodic function. 

 Thus,/(/) can be represented in terms of F(co) by the integral 



m = ^_j_^ F{c.)e'-^dc.. (5-2) 



TT. 



The functions /(/) and F{co) are often regarded as constituting a Fourier 

 transform pair which are mutually related by Equations 5-1 and 5-2. With 

 this terminology, Equation 5-1 is said to transform /(/) into the frequency 

 domain, while the operation indicated in Equation 5-2 constitutes the 

 inverse transformation. The symmetry of these transforming operations 

 is striking. 



As a concrete illustration of such a pair of functions, suppose that/(/) is 

 zero for negative values of time while for positive values it is a decaying 

 exponential: 



/(/) = e-', </< oo. (5-3) 



The spectrum is easily calculated: 



F{.^) = f (.-0 e-^-^dt = — ^- (5-4) 



In this case, the spectrum is complex. Upon performing the inverse 

 operation indicated by Equation 5-2, the exponeni.ial function given by 

 Equation 5-3 will again be obtained. We shall not carry out the details of 



