EXPANSION PROBLEMS. 385 



the general solution of the diflferential equation, when p 9^ 0, has 

 the form ^ 



Ae-"' -\- Be- "^"^ + Ce " '^'''^, 



where A, B, and C are constants. A solution which satisfies the first 

 two of the given boundary conditions for an arbitrary value of p is 

 obtained by setting A= 1, B = oi, C = or, and e>:cept for an arbitrary 

 constant factor this is the only solution which does so; in particular, 

 not more than one characteristic function can correspond to any one 

 characteristic value. A necessary and sufficient condition that 

 p 7^ be a characteristic value is that the solution just mentioned 

 take on the value when x = tt, that is, that p satisfy the equation 



(3) e- p"" + cue- ""^ + ore- "'"'^ = 0. 



As the problem is concerned at the outset only with the cube of p, 

 there is no loss of generality in supposing that p lies in the sector of the 

 complex plane for which 



(4) ~ 3 - ^''^ '^ - 3' 



For example, the equation (3) remains unchanged if p is replaced by 

 cop, the left-hand member being merely multiplied by co^. It wdll 

 accordingly be assumed henceforth that (4) is fulfilled, unless the 

 contrary is expressly stated. 



If the first term of (3) were absent, the equation would reduce to 



27rl 

 (5) g-a;p7r = _g 3 g- a;=pir^ 



which can be solved explicitly by taking the logarithms of both sides. 

 It has the infinitely many roots 



1 , 2n 



P n = 



3 V3 Vs' 



where n is any integer. Now if p is in the sector defined by (4), and 

 large numerically, the first term of (3) is very small, while at least 

 one of the others is large, and it is naturally conjectured, and readily 

 proved, that throughout that part of the sector which lies outside a 

 circle of sufficiently large radius about the origin the roots of (3) 



7 The general solution for p = is of course A + Bx + Cx^. It is obvious 

 that p = is not a characteristic value, since the insertion of this expression 

 in the boundary conditions gives at once A = B = C = 0. 



