531 Prof. T. H. Havelock. Some Cases of 
solutions will contain logarithms. Calculating the coefficients step by step, 
we obtain as a fundamental system 
1n = o8(1—hat geal gyalt ghtyat—...) a 
Ys = yi loga+a?(—8—$a+3a3— $5 a4+ ...) 
We know that Wj,; is a linear function of y and y2; however, it is simpler to 
obtain an expansion directly and use (60) to verify it. For this purpose we 
use the equivalent contour integral for the confluent hypergeometric function, 
ake a4 pet 1 (s) 1 (—s—k—m+ 3) (—s—k+m+}4) 
27t | _wi T(—k—m+3)T(—k+m+}) 
where the contour has loops if necessary, so that the poles of I'(s) and those 
of T'(—s—k—m+4)I'(—s—k+m-+4) are on opposite sides of it. The 
integral can be evaluated by the method of residues. When k = m = 1, the 
poles at which the residues have to be found are simple poles at s = —4, —3, 
together with double poles at s = 4, 3, §,.... The latter series gives rise to 
logarithmic residues. Carrying out the calculation, we obtain 
Wee = ads, (61) 
= = = 3 4p, 3 = DT (p+) 
= 1/2 of2 3/24 = 4-1/2) — 3/2 p—a]2 a SEEN UB 
Wii=7 ae (. +54 ) 4 arene log «Sry LDI(p 3) aP 
3 d I'(p+3) 
Yoo 2—5)+5 5 pce 0a Dna 2 
t4nt(y—2log2—5)+ 2 Cer @EH ©) 
where y is Euler’s constant 0°5772.... The coefficients may be put into 
alternative forms more suited for calculation ; for instance 
a T'(p+3) 
dp V(p+1)U(p+3) 
ont 13S pf aga il eit 
~ 2? .p!(p+2)! ae BEB MS) a ie 
For numerical calculation we have 
_ 3 -12,,3/2,-af2 4, 3 il 7 -B | 
Wasa gaues sare ga ee Vaan Tpa@ 7 
= (7+ log j a)(1+50+25 1 en att. } (63) 
° 4 192 
The expansion may be confirmed by comparison with the fundamental 
solutions of the differential equation given in (60); we find that 
(8/3) 74 Wii = (2 log 2—y¥—4) myo. 
For large values of « the general asymptotic expansion of W4, m is available ; 
and in this case we have 
Si. @ Wt . 1h 
~ —af2 55 SS } 
Wao (1435 +i iat) (64) 
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