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.

~91. 92.

Singularities.

155

the question arises, how is this value of to altered as the value S
changes. The value zo, chosen to begin with, can always be assumed
finite; for when the change of w from an infinite value z concerns us,
1
let us substitute s = -- a thereby converting 1) into a relation
between w and v', and investigate what values w assumes for z' =a.
Now we must first establish, what kind of singular points can
occur. An equation of the nth degree has always n roots. When
its coefficients are variable, as in the present case, these roots can
exhibit the peculiarities either of becoming infinite or of some of them
becoming equal. These two are the only kinds of singularities. As
we pursue the investigation, we shall show, that the number of such
points is finite.
1. Let us put wt = -, and determine the points at which w'= 0,
then we have instead of equation 1):
(F) Z) + Vw'i1() +.* * w'n-l -1.() + W'n(Pn() = 0.
If w' is to converge to zero, To(Q() must become = 0. This is an
equation determining k separate or coincident finite points for which
one value of w increases beyond any finite amount. These singular
points of the function to we shall denote by ac, 2....k; they are,
in the absence of further conditions, non-essential infinity points
(infinities); for, it follows from the equation
w P0 () = - p#(z) - (P(s) -..I A_ _)(z) - _ n (b),
that, though w is infinite for z = a, the product w q0(z) =- qP (z)
still remains finite.
In such a singular point, besides the one infinite value, - 1
further values of w, generally finite, will be found from the equation
wq"-91 (a) + -I- -2 2(a) +- * * * wpn-_ (a) -+ (pn (a) == 0.
Only when Spi(a) also vanishes, a second value of w becomes infinite;
when (p2(a) also = 0, a third; and so on. Such an infinity point a
can lose the non-essential character, because it is at the same time a
critical point. Critical points form the second kind of singularity.
2. To investigate an algebraic expression of the nth degree f(w)
at a determinate finite point wa, let us bring it to the form (~ 87):
f () = f (V-) + V-(w-tv)f (to ) + ( w1Z f"(w + ( '10 -
If f vanish simply for w = wl, we have f(zwl) = O, while the value
of its first derivate f'(wj) at this point is not zero. But if there be;X roots === wz, all the derived functions up to the (A - l)th inclusive
also vanish:
f/(ti) -=, /f(w1) =- 0,... /'-1 (/v) - 0.