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.

128

Functions of a complex variable.

Bk. II. ch. ii.

To all points of the plane z, which are outside the circle r = —,
correspond points in the plane a' within the circle = -. Circles
round the origin z = 0 change into circles round the origin z' - 0,
but these are described in the opposite direction; to the infinite in
the plane z corresponds only one point namely z' = 0. It is this
Circular Relation between two planes,*) called Transformation by reciprocal radii vectores or by Inversion, which we shall subsequently
employ in reference to infinite values of a.
80. When the values of a complex variable w = u + iv are so
determined by the values of a complex variable z = x + iy - rp, that
to each value of z within a determinate domain one or more values
of w can be assigned by means of any finite or infinite number of
arithmetical operations (~ 38) on z, w is said to be a function of
the complex variable g.
Here also functions are distinguished into one-valued and manyvalued, according to the number of values of tv belonging to one
value of z; into algebraic and transcendental, according to the
form in which the variables occur; and into explicit and implicit
according as the equation defining the function is solved for w or not.
The total course of a one-valued (monotropic) function is realised
by help of two planes. To each value x + iy = rT of the quantity z
corresponds a point of plane A having the rectangular coordinates x
and y, to each corresponding value t + -iv of zw, a point of plane B
having as rectangular coordinates it and v. If to each value of z
belong a determinate value of w changing continuously with z,
then to each point of plane A will correspond a point of plane B, to
each line a line, to each connected area a connected area. If on the
other hand, iv changes discontinuously at some points, while z changes
continuously, disconnected portions of plane B will correspond to a
connected area of plane A. In a word, the dependence of the quantity w
on z is geometrically represented as a Transformation of plane B upon
plane A.**) Such a transformation, for instance, was already investigated
in last Section by means of the equation: wv = -
81. Commencing with the case of an explicit function
wv = u -t iv = f(z) f= f(x + iy) = f { r (cos + — i sin P)}
-the quantities is and v are functions of the real variables x and y,
*) Mobius (1790-1868): Abhandl. der sachs. Gesellsch. d. Wissensch., 1855.
This paper on Circular Relationship (Kreisverwandtschaft) follows earlier notices
of the same subject in his Gesammelte Werke, vol. I1, p. 243.
**) Riemann: Grundlagen fiir eine allgemeine Theorie der Functionen.
Werke, p. 5.