660 BELL SYSTEM TECHNICAL JOURNAL 



functions. This would make it possible to shift without trouble 

 from any one expansion to any other expansion of the tabulation. 



I am under great obligations to my colleagues for their contributions 

 towards the preparation of this paper. I shall be grateful to any person 

 who will call. my attention to errors or omissions in any part of this 

 paper.® 



Notation 



The following notation is employed in Table I; also in Table II, 

 except as specifically restricted. 



a, b, c = positive reals. 



br :)c = branch X. For each multiple-valued function, branches 



are designated in one or more different ways. When 

 no branch designation is given, branch zero is to be 

 understood. 



C{z) = /^ cos(lTrz'')dz = - C(- z). C{± 00 ) = ± i 



cis(z) = cos z -\- i sin z = exp {iz) = e" = cisoidal oscillation if 



Z = lirft. 



Dy{z) = parabolic cylinder function of order v. 



Dn{z) = exp(-is2)ij„(s). D.^{z)={2ir)-h^K^{\z''). 

 D^,{z) = (ix)i expds^) erfc(2-^z). 



erf(z) =—= \ exp(- z^)dz = - erf(- z). erf(± «) = ±1. 



V77 Jo 



erfcCz) 



2 r* 

 = — p: I exp(— z''^)dz = 1 — erf (2). 



/ = frequency; parameter for the cisoidal oscillation. 



— 00 </ < 00. 

 ^(f) = coefficient for cisoidal oscillation, parameter/. 



Fn(J) = coefficient of an ^"-multiple pair (F„(/), i"Fn{g)) in 



pairs (223)-(225). 



« I am already much indebted to M. Paul Levy for a number of suggestions 

 including the expression of the general identical pair as the sum of any pair having 

 even coefficients and its transposed pair. 



