( 626 ) 
gy? cos20 == —1. 
In the same moment all terms in (X) become infinitesimal and 
the development of the series loses its significance. 
We will once more calculate tang—! « and inquire after the best 
value of a, always supposing that a must remain real. In the point 
zx—1 we have 
a—2 
— ’ 
uh 
Tite a 
and this mark not being able to fall below 1:a, the best develop- 
ment will be had for a=8. That series will be 
EE 
3 3 3 
1 2 4 8 1 Sh pea 4 8 16 
hale Crane ee tn 
1 2 4 8 16-32 
dlg tig tr tt =| + EE 
By writing it 
m= 0 1 
tang 1=1—2 = far z2)jn—ldz, 
nia 
0 
we easily get convinced of its accuracy. The terms written out give 
together 0.7854, that is in 4 decimals the value of tang! 1. 
The six first terms of the power series for tang Iz, 1. €. 
the only ones which were used here, would produce together 0.7440. 
The function log(1+2*) has the same singular points as tang! z. 
The region of convergence for the series of polynomials deduced 
from (LX) agrees with the one just considered. We have 
(ni (a—1)} = 
n= 
log (12°) = — = 
m=1m™ qm 
and for s=1 and a=3 we get 
1 
h=@ 1 Sos 
Age 
in oo Ch 1) 3241 te 
3 
5 ; : ih 
that is a series which really bears the mark ar 
6. Though after application of the simple auxiliary function 
(a—l)u 
g (u) = 
a—uU 
in many cases extensive regions of convergence can be obtained, 
as is evident from the preceding, after development of an arbitrary 
function F(x) there will generally be a limit to the extension of G). 
