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.

Non-essential singular point.

131

2. The rational fractional function: is one-valued
(z - a)m
and continuous in the entire plane, except at the non-essential singular
point z = a. The point infinity is not singular, since by the above
substitution we have for it: — ) = ( --- =
The most general rational fractional function is of the form:
Ca + a1z + a2z2 +. * a, n
bo + bz - + b2 - +. b mZ7
It will be shown in next Chapter, that every rational integer function
of the degree n may be broken up into n linear factors. If this result
be assumed here, and this fraction therefore written in the form:
6n (Z- a,) (b -- 2) * (Z - n)
b (Z-Pi1) (Z-P2). - (Z at)
where some of the quantities a as well as some of the f may be equal,
but each a will be supposed different from each (3 because otherwise
factors could be cancelled, then obviously: the rational fractional function
is one-valued and continuous in the entire finite plane except at the
non-essential singular points pf, f2... Pm. The point infinity likewise
is a non-essential singular point when n > m, but is a regular point
when n < m. For, the function behaves for -- co as:
an (21)(lo)~ **(,). n)~ an _(m —n(1- ) (l- 2Z) -.. (1 —CanX)
( i 1) (f 2 -2)(- ) bm (1-i') (1-Z') ( 1 —mZ
behaves for a' = 0.
3. The simplest explicit irrational algebraic function:
w = (' — a)T^, m a rational fractional number p- reduces to
the form gm by the substitution ' - a =. All propositions which
we shall prove for the function zm, can be easily transferred to the
more general form (' - a)m since we have only to observe that
whatever holds for the point z = 0 relates to the point ' = a.
If z = r (cos (p- i sin g), then for k = 0, 1,..q - 1,
w = i = ri (cos P ('p + 2kfc) + i sin q ( + 2a )).
This exposition shows that the function can be calculated. For,
both cosine and sine are series of powers. Each integer value of k
determines at each point the corresponding value of a branch of the
function; it is therefore a many-valued function, no longer singlevalued as those in the previous examples. At the point z = 0 and
at = o all branches have the same values, namely:
9*