PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 647 



desirable to give him coefficient pairs which are in closed form, that is, 

 involve only a finite number of operations with known functions. 

 Accordingly, the portion of the table expressible in closed form has 

 seemed to be the part which should be developed first. Specific pairs 

 requiring infinite series for their expression have not been included in 

 this preliminary draft of Table I. The omission of these series and 

 of other infinite processes does not signify any failure to appreciate 

 their importance. It is intended to include specific infinite series later. 



The Elementary Transformations of Coefficient Pairs 



The simple addition theorem (201) is of the greatest practical im- 

 portance. The summation may include any number of pairs; they 

 may be quite unrelated, or they may be the successive terms of power 

 expansions as shown in (106*)-(111*). Next to the addition theorem 

 we may place the multiplication theorem (202) or (203), special cases of 

 which are of great practical importance. Among these special cases 

 are (206)-(211) where any coefficient is multiplied or divided by its 

 parameter or by a cisoidal oscillation of its parameter. 



Any real linear substitution for the frequency and epoch parameters 

 is made possible by the simple transformations (205)-(207), (214). 

 The generalization of these transformations by the removal of the 

 restriction to real numbers is allowable in important cases as is 

 indicated by the parameters shown in square brackets with each 

 pair of Part 3. 



The differentiation and integration of coefficients with respect to 

 the frequency, epoch or other parameter give the important trans- 

 formations (208)-(213). 



Some of the simple transformations continue to yield new results 

 when they are repeated any number of times or when several trans- 

 formations are combined in sequence. Pairs (216), (218)-(222) are 

 examples of such combinations. All pairs in Parts 4-9 of this table 

 may apparently be derived from the seven key pairs of Part 3 by means 

 of these transformations employing complex parameters as indicated 

 in Part 3, and passage to a limit in certain cases. 



The resolution of pairs into the four types of ^"-multiple pairs, as 

 shown by pairs (223)-(225), throws considerable light on the nature 

 of coefficient pairs. 



Some of the elementary properties of pairs are expressed in words 

 as follows: 



Elementary Properties of Pairs 



(1) The sum or difference of pairs is a pair. Cf. pair (201). 



(2) Any constant multiple of a pair is also a pair. Cf. pair (204). 



