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.

126

Comnplex variable.

Bk. II. ch. I1.

Thus, for example, all complex numbers whose moduli are less
than rt and greater than r2 form a domain exhibited geometrically
by the plane ring bounded by the two concentric circles with radii r,
and r2 round the origin. But complex numbers whose modulus is
equal to rI form a linear figure, namely the circumference of the
circle whose radius is r1.
A domain of two dimensions is said to be connected, when we
can pass from any one point within it to any other without crossing
its boundary.
A domain of one dimension is said to be connected, when we
can pass from any point in the domain to any other point in it without
leaving the domain.
A quantity is said to be unrestrictedly variable within a domain,
when it is able to assume all numerical values belonging to this
domain.
A complex quantity is said to be continuously variable, when all
values which it assumes, belong always to a finite connected domain.
In particular a variable is called unrestrictedly continuous at a
certain point, when it can assume all values which belong to a
finite domain however small including that point. On the other hand,
the variable is restrictedly continuous at this point, when the values
it assumes near the point form a domain whose boundary passes
through the point, or form a domain of only one dimension. It is
discontinuous at this point when the point is isolated by itself, and
so belongs to no domain.
A further circumstance has to be noticed here: If a real variable
is said to alter continuously within an interval from a value a to
a value b, this informs us what numerical values it assumes, or in
geometric language, we know the path along which it travels. If a
complex quantity change continuously from a complex value a to a
complex value b, this tells nothing at all of its intervening values.
The ways in which it changes are just as illimitable as the continuous
lines which can be drawn joining one point of the plane to another.
Continuous change of a complex quantity z = x + iy requires
that both the real constituent x and the factor y of the imaginary,
vary continuously. A complex variable becomes infinitely small, when
its modulus becomes infinitely small, i.e. when both x and y have
zero as limit A complex quantity becomes infinitely great, when
its modulus becomes infinitely great, i.e. when either x or y or both
together increase numerically beyond any finite amount.
To the infinitely great values of complex numbers correspond in
the plane of the figure the infinitely distant points, and as a complex
number x + iy can become infinite, while the ratio x: y assumes all