484 MR. II. A. WEBB ON TUB OONtOtOBKOE 01? 



.connected area C in the 2-plane, within which p. n (z) is regular, though p. n (z) may have 

 singularities on the boundary of C. 



r is a positive quantity, and v//o(z) as well as/o( s ) must be regular within C. 



Then the approximate value of any solution of the equation (3) for large values 

 of n is 



y = A^fcJ.e^f^-^ + A^zJ.e-^'J^-* . . . . '. (4), 



where A t and A 2 are arbitrary constants, and <|i (2), \ji 3 (z), are functions of 2 that 

 remain finite as n increases indefinitely. 



This follows from Theorem I. 



Unless /* (2) possesses a line of singularities everywhere dense, forming a closed 

 curve, the result holds for all values of 2 except the singularities and branch-points 

 of/*(z). 



Denote the solutions whose approximate values are 



fc(*)^*faW'* and i/, 2 (2).<r l '" J 1^- d ' 



b y 



p n (z) and </ (2). 

 Denote 



Theorem II. The series 



(5), 



the c's being arbitrary save for the condition that the series 2 c n z* has unit radius of 



*ml 



convergence, is or is not absolutely convergent according as | 6 (x) \ is less than or is 

 greater than 1 (t) \ . 



xt 



If 1 6 (x) \ = j 9 (t ) | , F (x, t) converges or diverges according as S e converges or 



=i 



diverges. 



For the n th term of the series (5) is approximately equal to 



of which the modulus is 



whence the result. 



The examination of some special equations will illustrate Theorems I. and II. 

 3. For LAMP'S equation 



(6), 



y a* 



ik\BDZ*dz= + ilog (dnz + kcm), 



