132 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
IntRopucTORY ANALYSIS. 
(a) The Bessel function J is defined by the series 
m 
ali OO -5 pate) 
m2) = omtim\ 2-2m+2 2-4-2m+2-A4m+4 
For the function of the second kind we take as definition 
J m(Z) = esd (2) 
2 sin ma 
Gin(2) = 
This makes G,,z an analytic function of m, the value of which, when m is a real 
integer, 18 
Ee) = (Iog 2-y+ oN m\2) = Xaal(2) 
where Y,,(z) is Neumann’s function. 
In this case, therefore, Gz = —log2J,,z + a uniform function of z. 
In the following pages we are concerned chiefly with the function of order zero 
mt Li ae Be Ne eee INR 
or Cope ee oe )-te2(1 ~ 5+ ae go ae \-h+ (ts lee- ee 
When mod z is very large, while the phase (argument) of z lies between — 7/2 and 37/2, 
then approximately 
Ca et ei(e+z) e — 
: 22 
Similarly, when the phase of z is between 0 and 7 (excluding those values) 
mri my fy 
J 2 =e % pe) ead) 
ie 
(b) Ifw,y,zand p,,z are the rectangular and cylindrical coordinates of a point 
in space, so that ~=pcosw,y=psina, then the most important property of the Bessel 
Functions is that each of the eight functions 
(e@ or e-*) (J,,Kp or Grkp) (cosmw or sin mw) 
satisfies Laplace’s equation, or in other words is a potential function. 
Hence 
(v2+«")*(J,.Kp or G,,«p) (cosmw or sinmw)=0. 
Further, if 
RB? = (2-2) +y-y') 
= p? +p? — 2pp' cos (w — o') 
then 
(v? aF K?) . (JokR or G)«R) =()- 
Let now I= | | GoxR/(a’, y')da'dy’, the integral being taken over a finite area A, 
Then (y?+x2)I=0, if (x,y) is without, 
