Abel's theorem in problems and solutions - School of Mathematics

90 Chapter 2
study of analytic functions 17 .
FIGURE 36
2.11 Functions representable by radicals
DEFINITION. Let and be two multi-valued functions. By
we will denote the multi-valued function whose values at a point
are obtained by adding each value to each value of Similarly
one defines the functions and
By where is an arbitrary non-zero integer, we will denote
a function whose values at the point are obtained raising to power
each value
By where is a non-zero integer, we will denote the function
whose values at a point are obtained extracting all roots of order of
each value
311. Find all values of: a) b) c)
d) e)
DEFINITION. We will say that a function is representable by
radicals if it can be written in terms of the function and of
constant functions being any complex number) by means of
the operations of addition, subtraction, multiplication, division, raising
to an integer power and extraction of a root of integer order.
17 See, for example: Shabat B.V., (1992) Introduction to Complex Analysis, Pt. 2,
(AMS: Providence, R.I.).