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.

~ 82.

Branching section. Essential singular point.

135

I(-) the values r c (cos 2p + sin ),
II(-),,, 7 r (cos 4pr + i si n- ),
~~,, 5
II(-),,- rT (cos p i sin,
IV(-) r 5 (COS pn + i Sin 8),
vr/,^ - - 10-7r.. 1-g -l
V-),,,, r (cos -- + isin - =r5
We have therefore to connect Ip-) with I(+), I-) with III(),
III(-) with IV(+), IV(-) with V(+), finally V(-) with I(+), so that there
arises a connected five-leaved surface; every closed curve crosses the
branching section either not at all or a number of times that is a
multiple of 5.
If the exponent m be a real irrational number, the function is
infinitely many-valued; a single branch is constructed in the same manner
as before, moreover a Riemann's surface can also be formed but now
it must consist of infinitely many leaves.
4. The exponential function: ex+iy defined by its infinite
series, which has the same meaning as e (cosy + i sin y), is a onevalued and continuous function in the entire finite plane. But the
point infinity is a singular point, it is moreover an essential
singular point.
For, putting z =, and therefore:
1 1 '-iy',,
ez e= e'-iy' ex'-+2 ex22 (cos x'2+ y -isin 2Y2
x X j- -I" x'~.4-Y' 2
when the ratio y': x' of the vanishing values of x' and y' has any
arbitrary fixed limiting value 7 we have:
xk
x =0
The modulus of this expression converges to + oo or to zero according as x' approximates positively or negatively to the value zero,
while the functions cosine and sine oscillate between the limits - 1
and + 1. We can also make the values x' and y' converge to zero so
that the modulus may tend to any arbitrary finite value ek, by putting
Lim x,,,2 = k, for x' - O, y'= 0, and therefore x' = y'2k; in a
word: in the essential singular point, that for ez is situated in the point
S = ex, or for e3 in the point z = 0, the function is completely
indeterminate, it assumes every complex value without any restriction.