LQ Conway—The Partial Differential Equations of Mathematical Physics. 
Hence, putting 
Oe too de, 5 phos 
we arrive at 
1 T(u) du 
Qa }[(u = tp — PO 
where w is complex, and the integration is taken about a closed contour in the 
plane of wu. The result of the integration is, in general, zero, unless the contour 
encloses a pole of the subject of integration, such ag 
GStU—=f OF V=StbPe 
for +(m—1) is integral, and /(w) is taken to be a one-valued arbitrary function. 
Let us integrate along a contour enclosing the point w=¢—7r, and no pole of IK@)s 
then the result will be a finite series of negative powers of 7, of which the most 
important term, when 7 is small, is 
FE=9) 
and when + is large, 
where /“) denotes the n” derivative of /, it being supposed that this function and 
its derivatives do not become indefinitely large with +. 
It will be noticed that when the functions / are exponential or circular, we 
obtain a result proportional to the Bessel functions of order, an integer ++. The 
solutions thus arrived at are infinite when 7=0, 7.e. when 
R=0, MSO ...H =, 
without reference to the value of ¢; in fact, we get a “line” singularity, not a 
‘point ” singularity, in this space of (7 +1) dimensions. Another way of getting 
these results will be useful afterwards. We require a solution involving only r 
and ?. he differential equation may then be written 
n—-10p _ Hd 
od 
Or? a y or of. 
On substituting for 4, °F, we get 
Of OF (n—1)\(n—3) ,, _ 
BP iar UCN gee a Tae Om 
