1103 
ke} 
(wle) —a)m 
Sa) (U) = yn Soleil's ulm) (x) , 
mi! 
0 
a formula, whose correctness ensues at once from the ordinary 
theorem of Tarror for the theory of functions '). Evidently we have 
here a(@)=the maximum modulus of w (#)— on the circumference 
of a. If for the neighbourhood of «0, w(v)—w is a majorant 
function we have 
a (a) — w(a)—a 
consequently 
Si =o (0) 
All this holds endependently of the function to which S is to be 
applied. But the question, what is going to be the new domain of 
convergence does depend on it. For to that purpose the equation 
w (x) = red 
has to be solved for all values of 6 that are arguments of singular 
points of w on the circumference of (7) and to choose that solution 
for which w#—a, has the smallest modulus &; any circular domain 
round a, smaller than (&) in so far as it does not contain a singular 
point of the substitution function w(e) itself, is of such a nature 
that S ewists in the whole of that domain ; this domain therefore varies 
with the funetion considered. Thus for the transmutation (15), applied 
to the functions 
1 
i and). w= 
le? 14 
we have to solve resp. the equations 
ete, and ’+r=—1 
the former gives as radius of convergence of v= Tu the num- 
ber 4(VU5—1), in agreement with the fact that this w is a majorant 
function, the other the number 1, equal to the radius of convergence 
of the corresponding u itself. 
It may also occur that the transmuted of a function is not deter- 
mined in any neighbourhood of er, however small by a series 
of the form (1), though it ewists in all the points of such a 
neighbourhood. This happens when the circle of convergence of 
the funetion is smaller than the minimum circle (6), which still has 
the property that the series (1) produces a transmuted function in 
the point «, for all functions belonging to it. Take in the neigh- 
0 
bourhood of « =O 
1) Cf. No. 17. 
