194 Conway—The Partial Differential Equations of Mathematical Physics. 
Hence, we may begin by discussing the function A’, or 
© 2a 
re —t)—r} 
The path of integration may be deformed into (1) the real axis from — 0 to 
the point ¢—7—e, where ¢ is a small positive quantity; (2) a small circle of 
radius € surrounding the point ¢—7; (3) the real axis from ¢/—7—€ to —@. 
The integral (2) vanishes with «, and in (3) the subject of integration changes 
sign on account of the passage around the branch point. Hence 
Tf (ujda af Hee 
Ky = 2| : = = rv) ¢ 
oe Jom L@ = 6 =P 1 WVO= ll 
= Aene, oe opens | 
= 2 ; L4+yort sete... .|, 
since v> 1. 
Let us suppose /(¢—7v) does not become infinite for any real value of v 
from 0 to 0, and vanishes when v = 00; and that it can be expanded by Taylor’s 
theorem in positive integral powers of 7v, provided rv < S, where S is some 
positive quantity. Then 
I, Ree 2 [4= 0) ois) 2 [ F(t = rv) = f(a 
1 D) 0 v 0 (y 
: IL 
IO) =/C=®) =/ ( -- ) dv 
=I) 
The first integral is — f(¢)log7, and the second is a power-series in 7, and 
the third integral is a finite constant; for the subject of integration is always 
finite, and so is the range of integration. For example, when /(?) is e’, it becomes 
— ye’, where y is Euler’s constant. Hence 
= £(E = rv) d 
| LSE = — f(t) logr +a, 
1 
where a denotes a power-series in 7. 
In a similar manner 
= Sp Ghy 
1 pire PP ! 
[ S(t = rv)dv = f(b”? log r 
