166 
PHYSICS: P. S. EPSTEIN 
Proc. N. a. S. 
THE EVALUATION OF QUANTUM INTEGRALS 
By Paul S. Epstein • 
Norman Bridge Laboratory of Physics, Pasadena, California 
Communicated, May 24, 1922 
Rather a long time after its publication, my attention was drawn to 
a paper by Mr. E. C. Kemble^ dealing with the evaluation of the integral 
J = 2 dq, (1) 
a 
where a and b denote the roots of the radicand. Before preparing 
his own method of computation, Mr. Kemble criticizes two procedures 
used by other authors and rejects them as inadequate and even as errone- 
ous. This judgment cannot remain uncontradicted for in reality these 
procedures are both perfectly correct and very efficient. Since the in- 
tegral under consideration was first introduced into the theory of quanta 
by the writer of this note^ in order to formulate his quantum conditions 
for conditionally periodic motions, the moral obligation of putting things 
right is his. 
The first method to which Mr. Kemble objects is used when f{q) can 
be expressed in the form 
f(q) = <p{q) + a^Piq) (2) 
where (p{q) is quadratic in q, a denotes a constant and a\l/{q) is small 
compared with (p(q). The most obvious method consists in developing 
V/(g) = Q{cx) in powers of a 
Q{a) = Q(o) + aQ'io) + f Q"(o)+. ... (3) 
and integrating this series termwise. Mr. Kemble remarks that the in- 
tegrals of the higher terms cannot be calculated, "because the higher de- 
rivatives of V/(g) with respect to a become infinite dX q = a and q = b." 
It seems to me that the latter statement is not sufficient to draw such a 
conclusion; in the coefficients of series (3) there is, indeed, substituted 
0 for a, so that they remain finite at the values q = a and q = b, and so that 
the case requires a further investigation. That the roots of the function 
(p(q) can also in a certain sense be regarded as limits of integration can be 
deduced only if the method of complex integration is used, but the latter 
supplies at the same time the means for overcoming the difficulty of the 
integrand becoming infinite at those points. This method was given by 
Riemann^ just for the treatment of integrals of the type (1) and was first 
applied by Sommerfeld^ to problems of the theory of quanta. It is not 
necessary to expand upon it for it is explained with much detail in the 
