640 BELL SYSTEM TECHNICAL JOURNAL 



One form of integral is used both to analyze and to synthesize. 

 In both cases it is the product of the arbitrary function and 

 the elementary sinusoidal oscillation which is integrated. This 

 achieves the mathematical counterpart of spectrum analysis and 

 spectrum synthesis. The functions resulting from analysis and 

 synthesis stand in a mutually reciprocal relation.- They are paired 

 with each other. Either of these functions may be assigned with an 

 astonishing degree of arbitrariness. Singular cases being excepted, the 

 mate function is then determined uniquely and definitely by the 

 integral. While the sine, cosine and complex exponential are most 

 commonly used as the elementary expansion functions, an entire class 

 of functions present the same fundamental relations and find applica- 

 tions in the more recondite problems. 



The Fourier integral is an indispensable instrument in connection 

 with physical systems in which cause and effect are linearly related 

 (so that the principle of superposition holds) because it gives at once 

 an explicit formal solution of general problems in terms of the solution 

 for the sinusoidal case which is often readily found. This explicit 

 general solution makes use of two Fourier integrals, one for the spec- 

 trum analysis of the arbitrary cause and the other for the spectrum 

 synthesis of the component sinusoidal solutions. No further con- 

 sideration of the actual physical system is necessary after the ele- 

 mentary sinusoidal solution has been obtained. This point of view 

 has become a part of our general background of thought. 



Unfortunately the actual evaluation of specific Fourier integrals in 



closed form presents formidable if not insuperable difficulties. Only 



a small number of distinct general integrals have been evaluated in 



closed form in the century and more which has elapsed since the Fourier 



integral discovery was announced. Additions to the list of evaluated 



Fourier integrals can ordinarily be made only by the professional 



mathematician. Unless the physicist or technician is in a position to 



evaluate Fourier integrals by mechanical means, or is satisfied to 



employ infinite series or other infinite processes in place of the definite 



integrals, he is usually entirely dependent upon the evaluations which 



the professional mathematician has made in the past or is able to 



make for his special use. On this account, it is often desirable to so 



formulate practical problems that only evaluated Fourier integrals 



will occur. It would be well for the physicist and technician to become 



familiar with the Fourier integral evaluations which the professional 



mathematician has achieved. 



^ The fundamental importance of the Fourier integral may be associated with an 

 analogy which exists between the integral and the imaginary unit, both considered 

 as operators. In both cases two iterations of the operation merely change a sign 

 and four iterations completely restore the original function. 



