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.

The essential singular point.

339

Accordingly
() = aO+ f(x) F )+ f(Z) { F(z) }2+-* a. AZ t Fz)}n+**
F(z)=a0~a1l f(z) F(z)~2 -j-*
is an absolutely convergent series; for, its series of moduli:
AO + A A, R + A2 ()2 +.. Az () +
converges, because while r < B:
Lim +l.. < 1.
An
But as long as the series converges absolutely, it converges also
uniformly; in fact then, for every value of z, the amount of:
a {f(Z) } + an+rlf() } n+ +*
is smaller than A, provided we choose n large enough to make
An.n + A,,+lRn+l +.. < 6, remembering that B > abs f(V).
We have consequently:
Zj 2t Z.1 1 21.3)d3-a( cjds + a jcf(gd) dzC + C f()} 2 + anJ f(z) } d+ +
oo 20o o Zo0 O o
Since f(z) is a unique analytic function, the integrals on the
right are also analytic functions independent of the path of integration.
This series converges uniformly within its convergency,
Employing this theorem as a Lemma, we can find the value of:
-- z'2 + lz'l '' ~ ' + * 1' dz'
integrated round the point zero in a negative circuit, since the series
converges for every finite value of z' but zero. The value of this
integral is found by the substitution:
z = (cos q +- i sin (p),
to be:
2n
2 7i
f2 e-2,+ 1 e-3i' + 4e e-4T + n. — e-(,n+2) i -p+ p Xe+ y d&=O.
0
The essential singular point of the exponential function is therefore
not a branching point of the integral function; but it is an essential
singular point; for the integral:
-+ - 1 1+ ~
ae + 2 - +
is an infinite series of powers and has the singular point = 0.
22: