Confluent Hypergeometric Function. 131 



they find mathematical works of reference rather indigestible, 

 and gradually cease to consult them. For example, an 

 excellent summary, from a purely mathematical point of 

 view, of the properties of the function we are going to 

 consider is given in Whittaker and Watson's ' Modern 

 Analysis'*; but it would be hard reading for engineers. 



Or if it is objected that the engineer can hardly be 

 expected to be familiar with the function theory of linear 

 differential equations and may get into trouble over singu- 

 larities, he might reply, if sufficiently well read, that the 

 equation can't have singularities in the range of x considered, 

 unless f(x) or </>(#), or both, become infinite, and he would 

 notice that from the graphs. Or he might say that he is not 

 looking for a rule to which there are no exceptions. He 

 wants a rule that generally works quickly, and he is prepared 

 to risk an occasional failure, because he intends to refer the 

 finished design for a final check to an expert mathematician. 

 Divergent series have often been used by physicists in much 

 the same spirit, and with few, if any, failures. Finally, 

 many expert mathematicians have come into contact with 

 engineering work recently under war conditions ; they may 

 have opportunity and inclination to assist in design on the 

 lines we have indicated. 



§ 2. Properties of the confluent liy per geometric function. 

 We define the function M(a, 7, x) as follows : — 



M(«, h «,_l+ 1 _ iy .*+ 1#2 . 7(7 + 1) ^ +i.2.3. 7 ( 7 + 1Xy + 2)^ 



+ to infinity. ... (2) 



The series is absolutely and uniformly convergent for all 

 values of a, 7, and x, real or complex, except only when 7 is 

 zero or a negative integer ; this case is supposed to be 

 excluded. 



The function M(a, 7, x) has been discussed under various 

 notations by several writers f. The following is a list of 

 such properties of the function as are of use for our purpose; 

 most of them are easily verified from the definition (2). 



* Second edition, 1915, Chapter XVI. 



f For a list of references see Whittaker & Watson, he. cit. 



K 2 



