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.

374

Expansion of ambiguous analytic functions. Ilk. IV. ch. III.

have only non-essential singular points, either finite or at infinity, we
have the following theorem respecting ambiguous functions:')
When a function w has n values for each value of,, and in the
entire infinite plane its only irregular points are non-essential singular
points and branching points such as have been above discussed, the
function must be root of an algebraic equation:
f (z, W) 0,
of the nth degree in w, and of a degree m in z that is equal to the
sum of the orders (infinitudes) of the infinity points.
Let the n values of the function be denoted by w1, w,... n;
and further let axl, a2.. oa, be the finite singular points that are not
branching points; in each of these points one branch of the function
becomes infinite, let the orders of becoming infinite be denoted respectively
by i1, i2,...2 i- so that therefore the products:
V(Z - a)'; '( - ))i2;... WV(Z - CA-)Y
remain finite; where w in each signifies that branch which becomes
infinite in the point a involved. Moreover points a may be coincident.
Further let 1, 2,.. P. be the finite branching points that are
also infinity points; in these several points let kc, k2,... k leaves of the
function respectively be connected; and let the respective infinitudes
(cf. ~ 197) be denoted by 1t,,... Il; the products:
11 12 v1
W(Z - l)ki; W(s - 2)2;... ( -( )
are therefore finite. Lastly let the point z = oo be a branching point
in which k leaves are connected and let its infinitude be l, so that
to (ad )
is finite. Let us put
(it + i +... i,L) + (1I + 12 +.. V) + I - n.
Forming now the symmetric function of the values of w:
S = (a - W1) (a - w2)... ( - wa),
this, as in general every symmetric function of the quantities w, is a
single-valued function of z; for, even the paths along which certain
values of w interchange cyclically lead always to the same value of
the function S. S becomes infinite of the order i, in each point aoc
of the order l in each point f3B, and lastly of the order I in the point
z = oo. Thus as the single-valued function S has only non-essential
singularities, it must be a rational fractional function of z (~ 190)
that can be set down in the form:
*) Riemann: Theorie der Abel'schen Functionen, Werke (pp. 8t-135), p. 101.
Briot (j 1882) et Bouquet (t 1885): Theorie des fonctions elliptiques, 2. ed., p. 216.