An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

140

Functions of a complex variable.

Bk. 11. ch. IIT

and q (z) is a rational integer function; having chosen a lower limit
for n we can always determine 6 so as to make the difference on the
right less than any prescribed number, i. e.:
Lim mod [f(z 4 ) - f(z) = 0.
It is proved as in ~ 44, IV that this theorem of uniform or equable
convergence holds even on the limiting circle for a point in which the
infinite series converges, by varying z along a radius and so putting:
p (z) = aXn + 4 + *an + 1 il +4 1 n +t +
- 4 ( ) -a +)2 Zn + ai" + 6 n +l +
The amount of p ( - 6) is less than that of (7-T) M, where N is
put for the greatest amount in the series of the complex numbers:
an, a~zn + an, +1Cn+i, an + + an + +1+1 + an+2l + 2 etc..
For, choosing - 6 upon the radius to 2, z = Q is a positive
real quantity less than 1. Thus:
- (Z - ) =n —n,? w + Qn+ 1 an _n+ + Qn+2 Cn+2 + +..
and the powers of Q form a decreasing series that converges to zero.
But the Lem m a of Abel in ~ 44, IV can be stated as follows for
complex quantities.
If to, t,... t,... denote an infinite series of arbitrary complex
quantities and if the amount of the quantity:
Pm to + tl + * - ti,
is for all values of m always less than G, then the amount of:
r = 0oto + tl t +.. e tit,, is < G E,
when,, e1.. denote real positive decreasing numbers. For, we have
r o== 0 (0o - E1) + P1 (El 2) + N* p — 1 (8Em- - lm,) + PmrEm
as before, and so,
mod r < (o - Et) mod p + (I -,) modpl + — - (Em- - 8m) mod p,,_
+ E, mod p,n.
The numerical value on the right side is less than:
G ( +o -- El + el -,8 + * * *,ir --,III +,mI) G=. o.
84. The differential quotient of a function of a complex variable
at a point in which the function is continuous is formed as follows.
Supposing the complex variable - = x + iy to receive the increment
Az ==Ax +- iAy == Arei'p, let us consider the quotient of differences
f(z +xA) - f(z) or f(z + /r' ) - f'(z)
A Z Ar ei
The limiting values, to which the real and imaginary constituents of