588 
go?” 
and |rt 
2n von 
An + ay cm ee (3) 
These inequalities hold good, when we choose for v and n arbi- 
trary positive integers. If we take for » a fixed number, so that 
2" > p, there follows from (3) that | a_,| and | 6_,| are less than 
any positive amount. From this follows: 
Gn DL = 0, when 2” Sp: 
The expansion into a series of Log |F'(z)| contains therefore at 
most a finite number of terms with negative powers of 7. This holds 
also for the expansion into a series of the conjugate harmonical 
funetion 
9 2nv 
Arg. F (2) = AO + 2 (— bp cos nO + an sin nO) mm. 
For O< |z|<e we conclude that 
F(2) Sede. nn = ee 
in which w(z) does not have the point O as an essentially singular 
point; hence it has there either a pole or an accidental singularity. 
From (4) follows 
Of iw, OL . 
PG) Zz ee 
The function 1— F(z) satisfies the same conditions as /’(z). It 
is holomorphic in &, except in 0, and for |z| < 2¢ it is different 
from zero and 1. For this reason 
Pi BY vie 
F yl 7 eee OS leise = ae 
in which y(z) has the point either as a pole or as an accidental 
singularity. 
F(z)—1 
From (5) and (6) follows that TP has the point either as a 
| 2) 
pole or as an accidental singularity, so that the same holds 
for F'(z), whereby the theorem of Picarp has been proved. 
4. The non-essential extension of the theorem, which is as follows: 
“When F'(z) is meromorphic in a neighbourhood 2 of O with the 
exception of OQ and when in a neighbourhood of 0, 2’ < 2, il does 
not assume three values a, 6, and c, then F(z) is meromorphie in 
2”, appears directly, because the function 
F(z2)—a c—o 
‘F(z)—6b ee 
satisfies the same conditions as H’(z) above. 
Groningen, April 18, 1920. 
Pe (2) == 
