678 BELL SYSTEM TECHNICAL JOURNAL 



Essay a solution in the form of a power-series, multiplied by e-(i/2)x2. 

 f{x) = e-(i/2)x2 f; anx^\ (155) 



n=0 



Substitute this into the differential equation, and group all the terms 

 involving the same power of .r. For each such group, we have 



an+2{n + 1)(« + 2).x-" - a.(2« + 1 - C)x\ (156) 



and equating each group separately to zero, we arrive at the relation 



a„W«„ = (2w + 1 - C)l{n + \){n -f 2). (157) 



Put ao = 0, thus causing all the even-numbered coefficients to vanish ; 

 assign any arbitrary value to ai, and calculate the odd-numbered 

 coefficients as, a^,, Oy, and so onward. Or, put ai and all the odd- 

 numbered coefficients equal to zero, assign any arbitrary value to ao, 

 and calculate the even-numbered coefficients a-i, ai, a^, and so onward. 

 Either way we shall get a solution of (154), whatever the value of the 

 parameter C; but there are certain specific values of C which admit 

 a peculiar sort of solution. It is, in fact, evident from (156) that we 

 shall arrive at two entirely distinct results, according as C is or is not 

 equal to some value of {2n + 1) — according, that is to say, as C is 

 or is not an odd integer. For, if C is equal to an odd integer (2w +1), 

 the chain of coefficients will come to an abrupt end at the member 

 having that particular value of n; it and all the succeeding members 

 will be zero ; the power-series in the tentative (and adequate) expres- 

 sion (155) for the unknown function /(x) will consist of a finite number 

 of terms. But, if C is not equal to an odd integer, the power-series 

 will go on forever. 



Here we have a new kind of Eigenwert. If me parameter C, in the 

 differential equation for the curious kind of "string" which I have 

 just defined, has for its value one of the numbers: 



C = 2w + 1, w = 0, 1, 2, 3, 4, •••, (158) 



the equation enjoys a special sort of solution. If the parameter does 

 not have one of these Eigenwerte, the solution of the differential equa- 

 tion is altogether different. 



Let us see what difference these Eigenwerte make in the general 

 solution (155) of the differential equation. If the parameter C has 

 some other value than one of these, the series a„x" goes on forever; 

 and as x approaches infinity, the value of its summation increases at 

 such a rate as to overwhelm the steadily declining factor g-d/^)-^^^ so 

 that the function /(x) is infinite at both ends of the range — co <x< co . 



