~37~
Ci=s
Cq = glr[fl' - 8 -28] = 53~ 3,
Co = 2Hisa® = 38120 + 205 = 653" 4 653] = = 53 = 33 4 3
Co= 32 +35— 34
Co=28, 4 85=385~3,+3; .

Expressing these s's in terms of the ¢'s and wultiplying by

bt
T we get
sn 155 = q(sin 32 + 8in 2) + ¢®(sin 5z ~ sin 2) + ¢®(sin 72 - 2 sin 32 - sin 2)

+ ¢*(sin 92 - 2 sin 52 4 sin 3z 4 2 sin z) **ere

Expansion of functions requiring a convergence factor;
the Gamma function, and the sine function.

It ie known that the infinite product II[(1 + a'n,)‘flnv(x)]

is absclutely-uniformly convergent if (1 + a”x)‘Qflv(x) is taken as

: 1)V
a single factor and Qu(x) == apx + 4 a,%2% = o0 g L%L o'y,

and if v is chosen so that Slanl"" converges. ?¢

By exactly the same reasoning it can be proved that

Qflv (x)

[+ vy(x)} e ] is absolutely-uniformly convergent if

{1 + vu(x)} ¢Q"v(’) is taken as a single factor and

0 tass sV oy 2 2
Opy(x) == vy 4 4 v, see 4 TR and if v is chosen so that
Zlva|V** converges uniformly.

Suppose that the proper choice of v has been made. Then
Ony(x)

log {1 + v,,(x)} e has for its first term
<1 1 v
54)1 (1" = ( ) £ (@nt + Bx® 4 oga® 4 ores Y

When this is expanded by vhe multincmial theorem (see page 11),

every term contains powers of an, by, ¢4, *e**, such that the

——"fi—_‘_————-———-———.—_____._,______—_

*® Goursat, Vol. II, p. 127-1a1.