**10.
The general power w **= **z**^{a}*
*, where

Setting here *z = re*^{i}^{j}, we obtain Ln *z*
= ln *r + i*(*j *+ 2*k**p*) and, consequently,

where *k *is an arbitrary integer, whence
for *b *¹ 0 the function *z*^{a
}has always infinitely many
values which lie (for fixed *z* and *a*) on the circle |*w*|
=*r*_{k}* *with the radius

forming a geometric progression,
infinite on both sides, with the denominator *e*^{-2}^{pb}*. *The arguments of these values

form likewise on both sides
arithmetic progressions with the difference 2*pa*..

For *b *= 0, i.e., for
real *a*, the values *z*^{a}* *lie
on the circle |*w*| = *e*^{aln r}*
= r*^{a}* *with the arguments

If *a* = *p*/*q* is rational
(we assume that the fraction *p*/*q *has been
reduced), then all the values *q*_{k}*
*will differ from *q *among these values (for example,
*q*_{1 }*, q*_{2}*,
**···,**q*_{q-1})
by a multiple of 2*p**. *Consequently,
in this case, the function w = *z*^{a}*
*is finite-valued and coincides with the function ,

However, if *a* is an irrational real number, then
there are among the values *q*_{k}
in (5) those which do not differ from integral multiples of 2*p* and, consequently, the function *z*^{a}*
= e*^{aLnz }is infinitely-valued.

The multi-valuedness of the general power, like those
elementary functions considered above, is caused by the
multi-valued argument. The methods of their subdivision into
single-valued branches have been given above; the branch point is
*z* = 0.

Side by side with the general power, the function
(1) may be considered as the *general exponential function*

In contrast to (1), Formula (7) represents a set of individual
separate, not interconnected functions which differ by the factor
*e*^{2k}^{p }^{iz},
where* k *is an integer.

**1.****4 Integration of functions of a complex variable**** **We will consider
the concept of the integral of a function of a complex variable
and the most important properties of analytic functions,
connected with the concept of integral or operating on it. In
particular, for example, there will be established the
equivalence of the concept of analytic functions as functions
which are differentiable at every point of the region of their
definition as well as of the functions the integral of which does
not depend on the path (cf. Theorem 1 in **12.** and Theorem 3 in **17.**). This yields a new concept for the construction of a
theory of analytic functions. Applications of the concept of
integral and theorems, based on it, will be studied in the
following chapters.

**11.
Integral of a function of a complex variable****
**Let there be given some oriented curve *C* and on
it the function *f*(*z*) of a complex variable. By
definition, the integral of *f*(*z*) along *C *is

where *z*_{0} = *a, z*_{1}*,
··· , z*_{n+1}* = b *are
consecutive points which subdivide *C* into *n*
segments, *a *and *b* denoting the ends of *C*,
*z*_{k }an
arbitrary point on the segment |*z*_{k}, *z*_{k+1}|
of *C *and the limit is taken under the assumption that
max |*z*_{k+1}- *z*_{k}|
® 0.

If *C *is a sectionally
smooth curve and *f*(*z*) is a sectionally
continuous and bounded function, then the integral (1) exists.
The proof reduces to the theorem, known from analysis, regarding
the existence of the curvi-linear integral of a function of a
real variable *.

* Cf. Fikhtengoltz, Vol. III, p. 27 or Smirnov, Vol. II, p. 206 et seq.

In fact, setting

we find

The sums on the right hand side of (3) are integral sums for the corresponding curvi-linear integrals. Under our conditions, these integrals exist and, consequently, there exists

With the aid of (4), the calculation of the integral of a function of a complex variable is reduced to the calculation of real integrals.

Applying the introduced definitions, it is readily seen that
the derivative and integral of a complex function of a real
variable *w*(*t*) = *j**
*(*t*)*+ i**y*(*t*)* *is represented by the linear
combinations

Let *z* = *z*(*t*) = *x*(*t*)
+ *iy*(*t*) be a parametric representation of the
curve *C, *where *z*(*a*) *= a, z*(*b*)
= *b*; then, using (4), we reduce the calculation of the
integral of *f*(*z*) along *C* to the
calculation of the integral of the complex function of a real
variable

Formula (4) likewise yields that the usual properties of curvi-linear integrals transfer to integrals of a function of a complex variable:

(*a* and *b *being complex constants, *C*_{1}*
*+ *C*_{2}* *denoting the curve
comprising *C*_{1}* *and *C*_{2},*
C*^{-}* *the curve coincident with *C*,
but running in the opposite direction).

We will yet prove one property of the integral:

*Let M = *max* |f*(*z*)*| on the curve
C and l be the length of C, then*

The proof follows directly from the defined integral. In fact, we have

where is the length of the segmented line *z*_{0}*z*_{1}*
··· **z*_{n}*, *inscribed in the curve *C *and
in the limit, as max |D*z*_{k}|®0, we find (11).

**12. ****Cauchy's
theorem**** **In the general case, depends
on the integrand *f*(*z*) as well as on the curve *C*.
However, if *f*(*z*) is analytic in some simply
connected region, contained inside the curve *C*, then the
integral is completely determined by the positions of the ends of
*C* and does not depend on the shape of this line. In
other words, we have

**Theorem 1 **(A.Cauchy, 1825) *If the
function f*(*z*)* is analytic in the simply
connected region* *D*, *then for all curves C*,
*lying in this region and having common ends, the integral * *has
one and the same value.*

We will prove this theorem under the additional assumption *
concerning the continuity of the derivative *f* '(*z*)
(in the definition of analyticity in **5.**
was only required the existence of this derivative).

* For a complete proof, cf. Markushevich, [3], pp. 137 - 143, 154 - 262.

As always, let *f*(*z*) = *u*(*x,
y*) + *iv*(*x, y*); by the relation

(cf. (4) in **11.**),
the question of the independence of the integral on the
path reduces to that regarding the independence on the path of
the curvilinear integrals

However, as we know from analysis*, it is a
necessary and sufficient condition for the independence in a
simply connected region on the path of the curvi-linear integral where *P
*and Q are functions with continuous partial derivatives,
that the integrand of this integral is a total differential,
i.e., that at every point of the region *D* applies the
relation

* Cf. Fikhtengoltz, Vol. III, p. 68 or Smirnov, Vol. II, p. 216.

For the integrals (2), this relation becomes

however, the continuity of the partial derivatives follows
from the assumption of the continuity of *f '*(*z*).
Equation (3) coincides with the Laplace-Euler conditions and
satisfies them, since *f*(*z*) is an analytic
function, whence the theorem has been proved.

By this theorem for functions, analytic in simply connected domains, we can replace

by

where *z*_{0} and *z* denote the ends of
the curve *C*.

Based on Theorem 1, there may be proved several propositions, analogous to the usual ones of the integral calculus. First of all, there applies

**Theorem 2 ***If the function f*(*z*)
*is analytic in the simply connected region D, then the
integral*

*considered to depend on its upper limit, likewise is a
function analytic in D, where *

In fact, by the definition of the derivative and
the property of the integrals (9) and (10) of **11.**,
we have

By the strength of the continuity*, *f*(*z*)
may be written at the point *z*

where h(*z*) ® 0 as *z *® *z*;
substituting this into (6), we find

* The continuity of *f*(*z*)
is a consequence of its analyticity.

Since *f*(*z*)* *is constant for the
integration with respect to *z*,
we have

because it follows continuously from the
definition that Moreover, Inequality (11) of **11.****
**yields

(by Theorem 1, the path of integration from *z*
to *z + h *may be assumed to be straight, whence its
length is |*h*|). Thus, the first limit in (7) is equal to
*f*(*z*) and the second one equal to 0, i.e., *F*'(*z*)
= *f*(*z*), as was to be proved.

The function, the derivative of which is equal to
a given function *f*(*z*), is said to be its *primitive**. *The theorem
proved asserts that the integral of *f*(*z*),
considered as a function of its upper limit, is one of the
primitives of the function *f*(*z*).

**Theorem 3. ***Any two
primitives of one and the same function differ from each other
only by a constant term.*

Let *F*_{1}(*z*)
and *F*_{2}(*z*) be these two versions and

It is sufficient for the proof of the
theorem to show that *F*(*z*) is constant.

By the formula for the derivative (cf.
(15), **5.**), we have

because by our condition

whence

so that *u*(*x*, *y*) and *v*(*x*,
*y*) are constants and the theorem has been proved..

The following theorem allows to calculate integrals with the aid of primitives.

**Theorem 4. ***If F*(*z*) *is
an arbitrary primitive of the analytic function f*(*z*),
then

In fact, by Theorem 2, the function

is one of the primitives of *f*(*z*),
by assumption, so is *F*(*z*), whence by Theorem 3

where *C* is some constant. Setting here *z*
= *z*_{0}, we find *F*(*z*_{0})
+ *C* = 0, whence *C = - F*(*z*_{0}),
i.e., (8).

Note still that Cauchy's theorem, proved at the start of this section, may be given the form:

**Theorem ***If the function f*(*z*)*
is analytic in the simply connected region D, then its integral
along any closed curve C, lying in D, vanishes:*

The proof is based on the fact that the closed
contour *C* may be divided into two contours *C*_{1}*
*and *C*_{2}*,*with common starting and
end points (Fig. 18), By the properties of integrals

whence the vanishing of the integral along *C* is
equivalent to the equality between the integrals along *C*_{1}*
*and *C*_{2}*.*

Finally, we will prove one generalization of Cauchy's theorem which will be useful in the future. In fact, we spoke in this theorem in its last formulation about the integral along a contour, which lies entirely inside the region of analyticity of the function, while one has sometimes to consider integrals along curves on which the function, while remaining continuous, ceases to be analytic. It turns out that Cauchy's theorem remains valid also for this case:

**Theorem 5 ***If the function f*(*z*)
*is analytic in a simply connected domain D and continuous in
the closed domain ,
then the integral of f*(*z*), *taken along the
boundary* *C* *of this domain,* vanishes:

We assume at first that *C* is a "starry"
contour, i.e., there exists a point *z*_{0} such
that any ray with vertex at this point intersects *C* in
one and only one point. Without restricting the generality, it
may be assumed that *z*_{0 }= 0 (i.e., this means
a displacement of the *z*-plane), then the curve *C *may
be given the the equation *z* = *r*(*j*)*e*^{i}^{j}, where *r* is a
single-valued function. We denote by *C*_{l}* *the contour,
determined by the equation *z* = *l**z
= **l**r*(*j*)*e*^{i}^{j}, 0 < *l *< 1 (Fig. 19 below). Since *C*_{l} lies inside *D*,
by Cauchy's theorem,

However, as the point *z *describes
*C*_{l}, the
point *z* = *z */*l *describes
*C,* whence (10) may be rewritten

and, consequently,

Since the function *f*(*z*) is uniformly
continuous in * *(cf.
**5.**),
one may find for any *e *> 0
a *d* > 0, satisfying the
inequality |*z*-*z*|<*d *for any pair of points *z*,
*z *, whence

Let *l *be the length of the contour *C* and *R
=* max *r*(*j*);
with *l *> 1 - *d*/*R*;* *we will have
for any pair of points *z* and *z*

whence (12) will be fulfilled and we obtain by (11)

Since here *e *is
as small as we please and the integral does not depend on *e*, this integral is zero and the
theorem has been proved for "starry" contours.

Now, let *C* be an arbitrary sectionally
smooth curve. If *C* has sharp points, we remove from the
region *D* circles of small radius *e
*with centres at these points, so that the boundary of
the region *D*_{e}*
*obtained has no longer such points (Fig. 20). Producing
inside *D*_{e }a
finite number of lines *g*_{k
}(*k *= 1, 2, ··· , *m*), this region
may obviously be broken up into parts *D*_{k},
bounded by "starry" lines *C*_{k}*
*(*k* = 1, 2, ··· , *n*)*. By what has been
proved above, the integral along any line *C*_{k }vanishes:

* It is readily seen that a
segment of the sectionally smooth curve in a sufficiently small
neighbourhood of its points, not being a point of return, is
itself a "starry" curve. However, in the neighbourhoods
of the starry points, the curve may or may not be starry (for
example, the curve consisting of sections of the parabolas *y*
= *x*² and *y** = *2

We will assume that the lines *C*_{k
}continue in one, for example, positive
direction and add all the equations (13). Since for us every line
is travelled twice and in opposite directions, then all integrals
along *g*_{k }cancel each other (cf. (9) and (10) in **11.**).and the
remaining parts of the boundary *C*_{k}*
*form the boundary *C*_{e}* *of the region
*D*_{e}* *and,
consequently, the integral along this boundary vanishes:

There remains to show that the integral along the
boundary *C* of the region *D* vanishes; however,
this follows directly from the fact that *C* and *C*_{e}* *differ only
by a finite number of small arcs and since the function *f*(*z*)
is bounded, its integrals along these arcs likewise are small.
Thus, the integral along *C* differs as little as one
pleases from the integral along *C*_{e}*
*which vanishes, whence it is also equal to zero. This proves
the theorem completely.

**13
Extension to multiply connected domains**** **Cauchy's
theorem does not, generally speaking, apply to multi-connected
regions. In fact, the function *f*(*z*) = 1/*z *is
analytic everywhere in the ring 1/2 < |*z*| < 2, but
the integral from -1 to 1 along the upper and lower halves of the
circle |*z*| = 1 differ from each other. In fact, along
the upper semi-circle *C*_{1}, where *z* = *e*^{i}^{j}, 0<*j*<*p**,
*

while along the lower semi-circle *C*_{2},
where *z* = *e*^{i}^{j}, -*p*<*j*<0*,*

Hence we will adopt sometimes for the notation of
the integral from *a* to *b *along *C *in
multi-connected regions the symbol

However, if in a multi-connected region curves *C*_{1}
and *C*_{2} with common ends are such that they
bound one simply connected domain, belonging to *D*, then,
obviously, the integrals along such curves are equal, whence it
follows that *the value of an integral of an analytic function
in a multi-connected domain D is not changed, if the contour of
integration is deformed continuously so that its ends remain
fixed and it remains all the time in *D*.*

Let there be given in the multi-connected domain *D*
the points *a* and *b, *joined by the simple, i.e.,
without points of self-intersection, curve *C*_{0}*.
*Let *C* be any other curve, joining these points
(Fig. 21a). According only to this statement, one may,
without changing the value of the integral, deform the curve *C*
into another curve , lying in *D *consisting of :

1) the curve which together with *C*_{0 }bounds a
simply connected region, belonging to *D*;*
*2) the union of the simple closed curves

For the sake of convenience, we agree to denote by

We introduce the notation

for continuous deformation *g*_{k} , for which these curves remain inside *D,
*integrals (2) do not change, whence the quantities *G*_{k} are only determined by the function *f*(*z*)
and the domain *D*. Let *N*_{k}*
*be the integer which states the number and the direction of
the passage of *g*_{k} while constituting ; these numbers may be
positive, bounded or equal to zero (for example, in Fig. 21, *N*_{1
}= -3, *N*_{2 }= 1, *N*_{3 }=
0.) We find from this and the properties of the integrals (9) and
(10) in **11.****
**

The quantities *G*_{k} are are called *periods of the integral**
*of the function *f*(*z*) in the
multi-connected domain *D* or *cyclic
constant**s.*

**Example **Let *f*(*z*)
= 1/*z* and *D *be the "ring" 0 < |*z*|
< *R*, where *R* is arbitrarily large. Any path *C*
which joins the points 1 and *z* may be deformed as above
into the path , consisting of several passages of the unit circle |*z*|
= 1 and the simple line *C*_{0} joining the points
1 and *z* (Fig. 22).The integral along the circle,
counter-clockwise, when *z* = *e*^{i}^{j}* *and *j *grows from 0 to 2*p*,* *is

By (3),

where *k* is an integer, showing how many
times and in what direction the circle |*z*| = 1 has
contributed to the composition of *C *(in Fig. 22, *k*
= -2). By Theorem 4 of **12.**,

where ln denotes the value of the logarithm,
which is 0 at the point *z* = 1 and increases continuously
along *C*_{0}.

Assuming that *C* is an arbitrary path and
denoting the value of the integral of 1/*z* along it by Ln
*z*,* *Equation (5) yields

Thus, we have arrived again at the multi-valued function Ln z and explained its multi-valuedness from a new point of view.

Finally, we note that one may give the Cauchy
theorem of **12.****
**a somewhat new meaning so that it remains true also for
multi-connected domains. Let the function *f*(*z*)
be analytic in the multi-connected domain *D*, bounded by
the curves *C*_{1}, *C*_{2}, ···
, *C*_{n}* *(Fig. 23) and
continuous in *. *Introduce
the cuts *g*_{1},
··· , *g*_{n}*
*, converting *D *into a simply connected region *D*^{*}*,
*and denote by* C*^{*}* *the
boundary of this region - the curve consisting of the partial
curves *C*_{k}* *and the curves *g*_{k}, where the
last are travelled twice in the counter-clockwise directions
(indicated in Fig. 23 by arrows).

The function *f*(*z*) is analytic
in the simply connected domain *D** and continuous in ,whence,
by Theorem 5 of **12****.**
and the properties of Integrals (9) and (10) of **11.**

(the integrals along *g*_{k}*
*cancel each other and the remaining part along *C**
coincides with ). For this, we must assume that the curves *C*_{0}
and *C*_{1}, *C*_{2}, ··· , *C*_{n}
are covered so that the domain *D* remains all the time on
one side (for example, on the left hand side as in Fig. 23).
Thus, for regions of any connectivity, Cauchy's theorem holds in
the form:

**Theorem: ***If the
function f*(*z*) *is analytic in D and continuous
in , then its integral
along the boundary of this region , proceeding such that the
region D remains all the time on one side, is equal to zero.*

**14.
Cauchy's formula and the theorem of the mean **Let the
function *f*(*z*) be analytic in the *n*-connected
domain D and continuous in . We will show that there holds for any internal
point of this region Cauchy's formula (1831)

where *C* - the boundary of *D -* is travelled
in such a way that *D *remains all the time on the left
hand side.

We note that there enters on the right hand side of Cauchy's
formula only the value of *f*(*z*) on the boundary *C*
of the region. Thus, under the given conditions, the values of
the function inside the region are completely determined by its
values on the boundary: Cauchy's formula allows to evaluate the
function at any point of the region as soon as the boundary
values of this function are known.

For the derivation of Cauchy's formula, we remove from the
region *D** *a small circle of
radius

where the circle g_{r}
is travelled in the clockwise direction, whence

where g_{r }is
travelled in the counter-clockwise direction. We have on the
circle g_{r }that *z - *z = *re*^{i}^{j}, whence, since the
factor *f*(z) under the integral sign is constant with
respect to *z*, we find

By (2) and (3), we have

we now estimate this difference. By (11) of **11.**,

whence we see that our difference as *r* decreases may
be as small as we please. On the other hand, as is seen from the
left hand side of (4), this difference does not depend on *r*.
Consequently, the difference under consideration vanishes and
Cauchy's formula has been proved.

In particular, if the curve *C* is itself the circle |*z - **z*| = *R*, then,
setting *z - **z* = *Re*^{i}^{j}, we obtain Cauchy's
formula

This formula expresses the so-called *Theorem of the Mean *for
analytic functions:

**Theorem: ***If the function f*(*z*)
*is continuous in a closed circle and analytic inside this
circle, then its value at the centre of the circle equals the
mean of the arithmetic values on the circle.*

**15.
Maximum Principle and Schwarz's Lemma**** **We
will prove first the simple

**Lemma: ***If in some domain D*: 1) *the
real part of an analytic function f*(*z*) *or*
2)* its modulus is constant, then this function itself is
constant.*

For 1), the assertion follows directly from from the
d'Alembert - Euler condition: We have whence, by these
equations, also so that also We conclude therefore that *v*,
but that means that also *f*(*z*) is constant in *D*.

We will now give the proof of the Lemma under Condition 20.
Let |*f*(*z*)| º *M*,
where *M* is constant; for *M* = 0, the statement
is obvious. However, if *M* ¹
0, we consider the function *f*(*z*) = ln |*f*(*z*)|
+ *i *arg *f*(*z*), which in this case is
analytic. Its real part is constant ( = ln *M*), whence,
by what has already been proved, the function ln *f*(*z*)
is constant, which means that *f*(*z*) is constant,
and the Lemma has been proved.

We will now prove the *Maximum modulus principle *for
analytic functions.

**Theorem ***If the function f*(*z*)*,
not equal to a constant, is analytic in D and continuous in , then its modulus cannot
attain a maximum value at an internal point of D.*

By the properties of continuous functions (cf. **5.**),
|*f*(*z*)| attains its maximum* M* inside or
on the boundary of *D* (Fig. 24). In contradiction, we
assume that |*f*(*z*)| attains its maximum *M*
inside *D *and denote by* *the set of all
points of *D* for which |*f*(*z*)| = *M*,
i.e., |*f*(*z*)| is constant. It then follows from
the Lemma that also *f*(*z*) is constant in *D, *and
this contradicts the condition of the Theorem.

If * *does not coincide with *D*, then there
exists a boundary point* *z*_{0} of this set,
which is an internal point of *D*. By the continuity of *f*(*z*),
we have |*f*(*z*_{0})| = *M*,
because in any neighbourhood of *z*_{0 }is the
point . We construct the circle *C*: |*z-z*_{0}|=*r,
*belonging to *D, *so that on it there would be also a
point *z*_{1}, not belonging to
(this may always be done, because *z*_{0 }is a
boundary point of *). *Then* *|*f*(*z*_{1})|
< *M *and for any sufficiently small *e *> 0, by the continuity of *f*(*z*),
we may always say this of the point *z*_{1}as part
of *C*_{1}of the circle *C* on which

* Boundary and internal points of
a set are defined in the same way as for regions (cf. **3.**)

We denote by *C*_{2}* *the remaining
part of the circle; obviously, on it,

By the mean value theorem

where *ds *= *rd**j *is the length element of the
circle.

Going in (3) to absolute values and taking into consideration Inequalities (1) and (2), we find

where *l*_{1} and *l*_{2}*
*are the leng())ths of *C*_{1} and *C*_{2}*
*( *l*_{1} + *l*_{2}* *=
2*p**r*). However, the
last inequality if impossible, whence the principle has been
proved..

*NOTE If the function f*(*z*) *is
not constant, analytic in D and continuous in and, besides, does not
vanish, then also the minimum of |f*(*z*)*| cannot
be attained inside D. *

It is sufficient for the proof to apply the
maximum principle to teh function *g*(*z*) = 1/*f*(*z*).

The maximum principle of the modulus follows the useful for further applications

**Lemma (G.Schwarz** ***). ***If
the function f*(*z*)* is analytic in the circle*
|*z*| < 1 *and continuous in the closed circle,
where f*(0) = 0 *and if everywhere in the circle |f*(*z*)*|
*£ 1, then in the same circle

*For this, if there should be at one internal
point of the circle |f*(*z*)*| = |z|, then the last
equality holds in the entire circle and *

*where **a **is
a real constant.*

** Hermann Schwarz *1843 -
1921.

We consider for the proof the function

By** **the conditions of the Lemma, *j*(*z*) is analytic in the
ring 0 < *z* < 1 and continuous in the closed circle
|*z*| £ 1 (the continuity of
the point *z* = 0 follows from the fact that

We will prove in

**14.**),
we have, g 0 < |*z*| < 1 and continuous in the
closed circle |*z*| £ 1
(continuity at *z* = 0 follows from the fact that

We will prove in **22.****
**that from this follows the analyticity of *j*(*z*) at the point *z = *0;
thus, we apply to* j*(*z*)
the modulus maximum modulus principle. Since on the circle |*z*|
= 1, we have |*j*(*z*)|
= |*f*(*z*)/*z*| £
1; then also, by this principle, everywhere in the circle |*j*(*z*)| = £ 1, i.e., |*f*(*z*)| £ |*z|, *and the first part of the
Lemma is proved.

If now at every internal point |*f*(*z*_{0})|
= |*z*_{0}|, then at each point |*j*(*z*_{0})|=1;
however, then, by the maximum principle, |*j*(*z*)|º1 at all points of the circle and, by the
Lemma above, *j*(*z*) is
constant. Since |*j*(*z*)|
º 1, one may enter this constant in
the form *e*^{ia}* *,where* a *is
a real number; consequently,* f*(*z*)=*e*^{ia}*z,
*and Schwarz's Lemma is proved.

Geometrically, Schwarz' Lemma states that, for
any mapping of the unit circle onto a domain *D*,
lying inside the unit circle, with the aid of the analytic
function w =* f*(*z*), *f*(0) = 0, the image
of an arbitrary point *z *lies closer to the origin than
any point *z* (Fig. 25); however, if the image of anyone
point *z* were to lie at the same distance as the very
point, then *D* coincides with
the unit circle and the mapping is reduced to a rotation.

**16.
Uniform convergence**** **The following
work has an auxiliary character. We will consider the questions,
important for the sequel, of the uniform convergence of sequences
and series of analytic functions.

The sequence of functions *f*_{1}(*z*),
*f*_{2}(*z*), ··· is said to be *uniformly
convergent *to the function *f*(*z*) in the
domain *D *( or on the curve *C*) if for any *e* > 0 there exists a number *n*_{0}*,
*which depends only on* **e*,
such that there holds for *n *³
*n*_{0}, for all *z* of *D *(or on *C*)
the inequality

We will prove two theorems, analogous to the corresponding theorems of analysis.

**Theorem 1 ***The limit f*(*z*)
of the sequence of continuous functions *f*_{1}(*z*),
*f*_{2}(*z*), ···* f*_{n}(*z*)
*··· converges uniformly in some domain D *(*or on a
curve C*) *and is also a continuous function.*

We give a number *e *>
0 and denote by *z*_{0} an arbitrary point of the
region *D* (or the curve *C*). Due to the uniform
convergence, one can find a number *n* such that for all *z
*of* D *(on *C*)

By the continuity of *f*_{n}(*z*)
at the point *z*_{0}, one can find a number *d *> 0 such that for all *z*
of *D *(on *C*)

For such *z *and the *n* chosen
above, we have by the inequalities (2) and (3):

and this means the continuity of (*z*).

**Theorem 2 ***If the sequence of
continuous functions f*_{1}(*z*), *f*_{2}(*z*),
···* f*_{n}(*z*) *···
converges uniformly on the curve C to f*(*z*), *then
there holds the limit relation*

We give a number *e *>
0. By strength of the uniform convergence, we can find *n*_{0}*
*such that for all *n *³ *n*_{0
}and all *z* on *C*

where *l* is the length of *C*. For
such *n*,

which verifies the truth of (4).

The proved theorem allows to go to the limit under the integral sign in the case of uniform convergence of sequences of functions.

The concept of uniformly convergent sequences is closely
linked to that of uniformly convergent series. Functional series are said
to be *uniformly convergent *in the region *D* (or
on the curve *C*) if the sequence of its partial sums

converges uniformly in this domain (on this curve).

As in analysis, it proves to be convenient for applications for sufficient indications of uniform convergence of functional series.

**Theorem 3 ***If the functional series ** is
exceeded by some uniformly convergent numerical series i.e., if
for any point z of D*

*then the given functional series converges
uniformly in D.*

In fact, by a known comparison theorem, the given
series converges at any point *z* of *D. *Denote
this sum by *s*(*z*). In fact, any *n *remainder
*r*_{n}(*z*) = *s*(*z*)
- *s*_{n}(*z*) of this series
satisfies, by (5), the inequality

We have here on the right hand side the remainder
*r*_{n} of the converging numerical
series, which tends to zero as *n *®
¥, whence one may find for any *e* > 0 a number *n*_{0},
depending only on *e**, *beginning
with which there will be *r*_{n} < *e*, and then, by (6), for any *z*
in *D *and *n *³ *n*_{0},
one has the inequality

This as well is referred to as *uniform
convergence* of the given series.

By Theorems 1 and 2, the sum of a uniformly convergent series consisting of continuous functions is continuous and such a series may be integrated term by term, i.e.,

The question of the possibility of term by term
differentiation of a functional series will be studied in **19. ****(**Weierstrass'
theorem).

Consider now the family of functions *f*(*z*,
*a*), depending on a (real or complex) parameter *a*.
One says that *f*(*z*,* a*) tends for *a*®*a*_{0}
to the function *f*(*z*)* *with
respect to* z *in the domain *D *(or on the curve *C*)
if for any *e **> *0
can be found *d = d*(*e*) such
that for |*a* - *a*_{0}| > *d** *for all *z* of *D*
(or on *C*) one has the inequality

In the same way, one can prove for sequences that the limit of a uniformly convergent family of of continuous functions is a continuous function and that for such a family there holds the limit relation

In the sequel, we will have to deal with
integrals along unbounded curves - *improper
integrals*. For this, we will always consider only such
curves *C*, segments of which belonging to an arbitrary
circle are sectionally smooth. The function *f*(*z*),
given on *C*, will be assumed to be sectionally continuous
and bounded.

We will now determine the integral of *f*(*z*)
along an unbounded curve *C*. To start with, let *C*
be unbounded only on one side and *a* be its end. We will
then denote by *C *the part of *C* with end* a *and
of length* l *and set by definition

where, if this limit exists, we will say that the
(improper) integral *converges*. If *C* is
unbounded in both directions, we define the integral as the sum
of integrals of two parts, on which *C* is subdivided by
an arbitrary point *a*.

Let *f*(*x, **z*)
be defined for all *z* in *D *and for all *z *on the line *C*. We will
say that the integral

*converges uniformly *in *D*, if
there exists for any *e *> 0
a number *l*_{0} such that for all *z* of *D
*for any *l > l*_{0}

(we assume that *C* is unbounded on one
side; the extension to the general case is done as above).

**Theorem 4 ***If the function f*(*z,
**z*) *is analytic with
respect to z and sectionally continuous with respect to **z **for all z in the simply
connected region D and for all **z **on
the line C and the integral*

*converges uniformly in D, then it is analytic
in this region *(cf. Theorem 1 in **19.**)

In order to prove this theorem, we employ the
theorem, inverse to Cauchy's theorem, according to which the
function *F*(*z*) is analytic in the simply
connected domain *D*, if it is continuous in this domain
and its integral along any closed curve, belonging to the domain,
vanishes (for the proof of this theorem refer to **16.**).

Under the conditions of the proved theorem, the
continuity of the function *F*(*z*) is established
as usual (as Theorem 2 of Relation (9)). There remains to prove
that the integral of *F*(*z*) along the arbitrary
closed contour *G* , belonging
to* **D* , vanishes. We
have

By the uniform convergence of (12), according to a theorem, known from analysis*, one may interchange the order of integration and obtain

since the inner integral vanishes, by Cauchy's theorem. This proves Theorem 4.

** Cf. Fichtengoltz, Vol. II, p. 733: what is stated there relates to real definite integrals, but after introduction of a parameter and separation of real and imginary parts Integral (13) is reduced to such integrals.

Note that in the case of an unbounded curve *C*,
it is unnecessary to make for the analyticity of the function *F*(*z*)
any additional assumptions regarding the convergence of the
integral (12); this follows from the possibility of interchanging
without further assumptions the order of integration in (13).

**17.
Higher order derivatives**** **By
definition, an analytic function - a function of a complex
variable - has a derivative at every point of a certain domain *D*
(cf. **5.**)
We will show that there follows automatically from the
analyticity of a function the existence and analyticity of all
its consecutive derivatives.

**Theorem 1 **(A. Cauchy 1842) *If
the function f*(*z*) is analytic in *D and
continuous in *, *then
it has at every point of D derivatives of all orders, where the
n-th derivative is given by*

*where C is the boundary of D.*

Let *z *be an arbitrary internal point of *D.
*By the definition of the derivative and Cauchy's formula in **14.**,
which we apply for the points *z* and *z + h*, we
have

However, obviously, as *h *®
0, the function 1/(*z - **z
- h*) tends for all *z *on *C
*to 1/(*z - **z*)
and, consequently, by Theorem 2 of **16.**** **(for
the case of a set of functions, depending on the parameter *h*),
the limit exists, where

The theorem has been proved for *n *= 1.
Assuming that it holds for any *n* - 1, one can prove in
exactly the same way its truth for *n *and thereby prove
the theorem completely.

**Note 1. **As is seen from the
proof, the theorem may yet be formulated as follows: *If the
function f*(*z*)* is
continuous on the boundary C of the domain D, then the function*

*representing Cauchy's formula, is analytic in this region.*

**Note 2.** Equation (1) for the derivatives is
obtained formally by differentiation of Cauchy's formula with
respect to *z*; the theorem proved confirms the
correctness of this differentiation.

Equation (1) yields the important *Cauchy
inequality*. Denote by *M* the maximum modulus of *f*(*z*)
in the domain *D, *by* R *the distance of the point
*z* from the boundary *D *and by *l *the
length of this boundary. Equation (1) yields

In particular, if *f*(*z*) is
analytic in the circle |*z* - *z*_{0}| <
*R*, then, using this circle in the capacity of *D*,
we obtain

which is Cauchy's inequality.

We will employ these results for the proof of two important theorems of the theory of analytic functions.

**Theorem 2 **(A.Cauchy, J. Liouville)* *If
the function f*(*z*) *is analytic throughout the
plane and bounded, then it is constant.*

* Th theorem was first proved by Cauhy in 1844, but used in an essential way in the work of J. Liouville (1809 - 1882).

Let everywhere |*f*(*z*)| £
*M*. For arbitrary points *z* of the plane and for
any *r*, Inequality (5) yields for *n* = 1

Since here the left hand side does not depend on *r
*and the right hand side may for increasing *R* be
made as small as desired, one has |*f* '(*z*)| = 0.
Thus, *f* '(*z*) º 0,
whence, by Theorem 4 of **12.**,

i.e., the function *f*(*z*) is constant and the
theorem is proved.

**Note **Theorem 2 admits the generalization:

If the function *f*(*z*) is analytic throughout
the plane and its modulus does not increase faster than *M*|*z*^{n}|,
where *n* is constant, then this function is a polynomial
of degree not higher than *n***.

** In particular, we obtain for
*n* = 0 Theorem 2.

The proof is analogous to the
preceding one: Let *z*_{0} be an arbitrary point
of the plane; we have by Inequality (5)

and, noting that
here |*z*| £ |*z*_{0}| + *R*,
we find after going to the limit *R *®
¥ that *f *^{(n +
1)}(*z*_{0})
º 0. Since *z*_{0} is an arbitrary point of
the plane, then *f *^{(n + 1)}(*z*)
º 0, whence, by the same method as
above, one can arrive at the result required.

The following theorem is the inverse of Cauchy's
theorem in **12.**

**Theorem 3 **(G. Morera (1856 -
1909) 1886) *If the function f*(*z*) *is
continuous in the simply connected domain D and the integral along
any closed contour in D vanishes, then f*(*z*) *is
analytic in this domain.*

The theorem states that in *D *the
integral does not depend on the path of integration, i.e., it
determines for fixed *z*_{0}*
*a certain function of *z*:

Reverting literally to the proof of Theorem 2 in **12.**, we see
that this function has the derivative *F* '(*z*) = *f*(*z*),
i.e., it is analytic (in this theorem, we have used only the
continuity of *f*(*z*) and the independence of the
integral on the path). However, by Theorem 1 above, *f*(*z*)
as the derivative of an analytic function is on its own merit an
analytic function. Thus, Morera's Theorem has been proved.

**1.****5. Representation of analytic
functions by series**** **We
will consider the problem of the representation of analytic
functions by power series and their generalization - series of
positive and negative powers of (*z* - *a*)*. *The
expansion of functions in series is not only of theoretical, but
also of practical interest. For example, we say that with the aid
of series we may compute approximate values of a function; in
many problems of an applied character (solution of differential
equations, etc.), solutions are obtained directly in the form of
series.

We will restrict consideration
here to the basic theoretical aspects, linked to expansion of
functions in series; the majority of them will have a very
important role in the following exposition of the theory of
functions of a complex variable and its applications (for
example, especially in Chapter V et seq.). In particular, there
will be established the equivalence of the concepts of analytic
functions (in the sense of **2.**) as well as of functions ,
differentiable at every point of the region of definition and
also of functions, differentiable at every point of their region
of definition and also of functions, representable in the
neighbourhood of every point as sums of power series (cf.
Taylor's Theorem in **18. **and Theorem 3 in **19.**);
this presents yet another concept of the theory of analytic
functions. In **25**, we generalize the concept of
analyticity by extending it to multi-valued functions.

**18.
Taylor Series****
**We begin with a generalization of formulae, known from
analysis as Taylor's formulae, and prove on its basis that every
function, analytic at a point, is represented in the
neighbourhood of this point by the sum of a power series.

We will employ the formula for the sum of the terms of a geometric progression

by rewriting it in the form

(this formula is also true for complex *q*).
Fixing some point *a* in the domain *D *of
analyticity of *f*(*z*) and using (1), we write

Multiply now both sides of the equation by (1/2*p**i*)*f*(*z*) and integrate it with respect to*z *along some closed contour *C*,
lying in *F* and containing *z *and *a.*Using
Cauchy's formula in **14.**
and the formulae for their derivatives in **17.**, we
obtain *Taylor's classical formula**

where the remainder term has the form

There arises the question under what conditions *R
*® 0 as *n *® ¥ or, what is the same
thing, under what conditions the function *f *(*z*)
is represented by its* Taylor expansion* with centre at
the point *a*, i.e.,

The answer to this question is given by the

**Theorem **(A.
Cauchy, 1831) *The function f*(*z*)* is
represented by its Taylor series *(*4*)* in any
open circle with centre at the point a, in which it is analytic.
In every closed region, belonging to this circle, Taylor's
expansion converges uniformly.*

*** An expansion of sucha
form (for real *z)* is first encountered in a paper in
1715 by Brook Taylor (1685 - 1731), but its systematic
application began only in 1742 in the work of Colim Marlaurin
(1698 - 1746).

Let *R* denote the radius
of the circle of analyticity of the function *f*(*z*)
(with centre at the point *a*) and consider an arbitrary
number *R*', 0 < *R*' < *R, *and the
circle |*z - a*| £ *kR'*,
where *k* < 1 is an arbitrary positive number. Let *z*
be any point of this circle and *C* the circle |*z - **a*| £
*R'*. We have |*z - a*| £ *kR'*,
|*z - **a*| £ *R'*, whence

and (3) yields

where *M *is the maximum modulus of *f*(*z*)
in the circle |*z - a*| £ *R'* (the function *f*(*z*)
is analytic in this circle, whence it is bounded). Since *k*
< 1, this shows that *R*_{n}* *® 0 as *n *®
¥, where the estimate of *R*_{n
}does not depend on *z*; thus, in any circle |*z
- a*| < *kR*', where 0 < *k* < 1, the
Taylor series converges uniformly.

We may embed an arbitrary closed region, lying in
the circle of analyticity of the function *f*(*z*)
in some circle |*z* - a| < *kR'*, where 0 < *k*
< 1, whence also in such a region the series converges
uniformly, and the theorem has been proved.

Thus, every function, analytic in a circle, may
be expanded in a power series. There arises the question whether
conversely an analytic function is the sum of an arbitrarily
converging power series? In order to answer this question, we
must consider certain properties of power series. This will be
done in **19.** We
now present the Taylor expansions of certain elementary
functions:

they have been written down for those
single-valued branches which are equal to 0 and 1, respectively,
for *z* = 0. The methods for obtaining these expansions
are the same as in ordinary analysis.

* In this formula, *a* is
an arbitrary complex number; in the particular case of natural *a*
= *n*, the series ends at the *n*th term and,
consequently, converges in the entire plane.

We start with two general theorems relating to uniformly converging series, comprising analytic functions ; these theorems were first proved by K. Weierstrass in 1859.

The first of them shows that uniform passage to the limit preserves the property of analyticity.

**Theorem 1 ***If the series*

*consisting of functions which are analytic in
a simply connected domain D, is uniformly convergent in this
domain, then its sum likewise is a function which is analytic in
D.*

In fact, according to **16.**** **,
the sum *s*(*z*) of Series (1) is continuous in *D.
*Let *C* be an arbitrary closed contour, lying in *D*;
by the uniform convergence of Series (1), this may be integrated
term by term along *C *to yield

because, by Cauchy's Theorem in **12****.**,
the integral of analytic functions *f*_{n}(*z*)
along a closed contour in a simply connected domain vanishes.
Now, by Morera's Theorem of **17.**, we
may assert that the function *s*(*z*) is analytic
in the region *D, *whence the theorem has been proved.

The second theorem shows that for analytic functions the question of the possibility of term by term differentiation of the series is resolved more simply than in ordinary analysis.

**Theorem 2 ***The arbitrary
series *(1), *consisting of functions analytic in the
domain D and continuous in , uniformly converging in , may be
obtained by differentiation any number of times in D.*

Let *z *be an
arbitrary point on the boundary *C* of the domain *D, *z
an arbitrary internal point of *D*. Since the difference *z - **z *is bounded for fixed
*z *below by the modulus of a positive number, the series where *k*
is an arbitrary natural number, converges uniformly with respect
to *z *on *C*.
Consequently, it may be integrated term by term along *C, *whence
converges the series

(for every term of the series, we may use
Cauchy's formula for the derivatives in **17.****
. **It remains to prove that the sum of Series (2) is the *k*-th
derivative of the sum *s*(*z*) of Series (1).
However, by the uniform convergence of the left hand side, one
may write (2) in the form

(we employ again the same Cauchy formula), as was required.

**Note 1. **In order to assert the
uniform convergence of a series of analytic functions in a closed
domain *, *it is sufficient to require its uniform
convergence on the boundary of this region. This follows directly
from the maximum principle of **15.**,
according to which

**Note 2. **A simple example shows
that one may assert in Theorem 2 the convergence of the series
from the derivatives only in the domain *D*, and not in *. *In fact,
obviously, the series* *converges uniformly in the closed
cir*cle |*z*| *£ 1, because it becomes stronger
than the convergent numerical series However, the derivative
of the series (convergent by Theorem 2 for |*z*| < 1)
diverges at the point *z* = 1 of the boundary of the
circle.

In the sequel, the dominant role belongs to power series. The character of their convergence is explained by

**Theorem 3 **(N.
Abel 1826 (1802 - 1829)) *If the power series converges
at the point z*_{0}, *then it converges also at
any point z, located closer to the centre a than z*_{0}*
, where in any circle |z - a| *£ k*|z*_{0}*
- a|, where *0* < k < *1* the series
converges uniformly.*

We assume that *z* is an
arbitrary point of the last circle and represent the *n-*th
term of the series in the form

By the convergence of the series at the point *z*_{0}*, its general term tends to
zero and, consequently, is rounded at this point, i.e., |c*_{n}(*z*_{0}*
- a*)^{n}*|**
*£
*M *for all *n*. Besides, by
assumption, |(*z - a*)/(*z*_{0}* - a*)|
£ *k*, whence for all *n
*

This yields the uniform
convergence of the series in the circle |*z - a*| £ *k* |*z*_{0}*
- a*| and finally the proof of Abel's theorem.

By Abel's Theorem, the region of convergence of
the power series is* *the open circle with
centre at the point *a *(which might likewise degenerate
into a point or occupy the entire plane) and yet, may be, be
several points on the boundary of the circle. The radius of this
circle is referred to as the *radius of
convergence* of the power series.

The formula for the determination of the radius of convergence is

where denotes the upper limit (Fichtengoltz, Vol. I, p.
107). The formula was obtained by A. Cauchy in 1821 and found
essential use by J. Hadamard (in the 20-th Century). It is
referred to as the *formula
of Cauchy-Hadamard**.*

For its derivation one must prove that for any *z* for
which |*z - a*| £ *kR*, the
power series converges and for any *z*, for which |*z-a*| >* R, *this series
diverges. By definition of the upper limit, for any *e *> 0, one finds *n*_{0}*
*beginning with which

We choose *e *such that

when for *n* ³ *n*_{0}* *and |*z
- a*| £ *kR*

Since 2*k*/(*k* + 1)
< 1, by a known theorem of comparison of series, formed by the
terms on the left hand side, it converges.

Moreover, the determination of the
upper limit yields that for any *e *> 0 an infinite
sequence *n* = *n*_{k}* *for
which

However, for |*z* - *a*| > *R,*
one may always choose *e* so that (1/*R* - *e*)|*z -
a*| > 1, when for our sequence *n* = *n*_{k}*,
*corresponding to this*
e** *, the term will
increase unboundedly and, consequently, the power series will
diverge (its general term not tend to zero).

The theorems of Weierstrass and
Abel yield an affirmative answer to the question in **18.**

**Theorem 4 ***The
sum of any power series in its circle of convergence is an
analytic function.*

In fact, let |*z - a*| <
*R* be the circle of convergence of our power series. In
any circle |*z - a*| £ *kR, *where 0 < *k *< 1, the
convergence is uniform by Abel's Theorem and, since the terms of
the series *c*_{n}(*z* - *a*)^{n}
are analytic functions, then, by Weierstrass' Theorem, this sum
is analytic in this circle. However, since any internal point *z*
of the circle of convergence may be enclosed in some circle |*z*
- *a*| < *kR*, where 0 < *k* < 1,
then has been proved thereby the analyticity of the sum of the
series in the entire circle of its convergence.

Finally, we prove the truth of the

**Theorem 5 ***Any power series is the Taylor
series of its sum.*

In fact, let in some circle

Setting here *z* = *a*, we find *f*(*a*)
= *c*_{0}. Differentiating (5) term by term and
then setting *z* = *a*, we find *f* '(*a*)
= *c*_{1}. By consecutive differentiations and
then setting *z* = *a* we find

whence

and Series (5) actually is the Taylor series of *f*(*z*).

Theorem 5, referred to as the *Theorem of the
uniqueness *of the Taylor series expansion, because it
follows from it that having found by any method an expansion of
an analytic function *f*(*z*) in a power series, it
is the Taylor expansion of this function.

Moreover, one may conclude from this theorem and
the theorems of **18.****
**that the radius of convergence of the power series (5)
coincides with the distance from the centre *a *to the
closest points at which the analyticity of the sum *f*(*z*)
of this series is violated. For example, the radius of
convergence of Series (6) in **18. **equals 1.
because for *z* = -1 their sums lose their analyticity
(naturally, we consider the second series when *a* is not
an integer).

**20.
Uniqueness Theorem **We have seen in **14.****
**that an analytic function is determined completely by
its values on the boundary of its domain of analyicity. We will
show here, in agreement with this, that an analytic function is
determined completely by its values by its values on an arbitrary
sequence of points, converging to some internal point of the
region of analyticity.

We will start with a theorem relating to the
zeroes of an analytic function. Any point *z* = *a *at
which a function *f*(*z*) vanishes, i.e., at which *f*(*a*)
= 0, is called a *zero* of *f*(*z*).
If an analytic function is not identically 0 in the neighbourhood
of its zero *a*, then in its Taylor series with centre at
a not all coefficients may be zero (otherwise, the sum of the
series would be identically equal to zero). The number of the
least different from zero coefficient of this expansion is called
the *order *of* *the zero of *a. *Thus, in
the neighbourhood of a zero of order *n*, the function's
Taylor expansion has the form

where *c*_{n}* *¹ 0 and *n *³
1.

Obviously, one may *determine the order of a
zero a from the order of the least non-zero derivative f *^{(n)}(*a*).

Likewise it is obvious that in the neighbourhood
of a zero of order *n* an analytic function may be written
in the form

where the function

is likewise analytic in the neighbourhood of the
point *a* (because it is represented by a convergent power
series).

By the continuity of *j*(*z*),
this function differs from 0 also everywhere in some
neighbourhood of the point *a*, whence follows

**Theorem 1 ***Let the function f*(*z*)
*be analytic in the neighbourhood of its zero a and neither
equal identically zero nor sero anywhere in its neighbourhood.
Then, there exists a neighbourhood of the point a in which f*(*z*)
*does not have zeroes other than a*.

From the proved theorem follows the *uniqueness
theorem *of the theory of analytic functions which we will
discuss in **21.**

**Theorem 2 ***If the functions f*_{1}(*z*)
*and* *f*_{2}(*z*) *are analytic
in the domain D and their values coincide on some sequence of
points a*_{n}*, converging to an internal
point a of D, then everywhere in D*

For the proof, we consider the function

It is analytic in *D,* has its zero points
*a*_{n}, and by its continuity also the
point *a*, because

whence *f*(z) is identically zero in some
neighbourhood of *a*, because otherwise it would violate
only the proved Theorem 1. Thus, the set of all zeroes of *f*(*z*)
would be a sole internal point.

Denote by the set of all internal points of the set
of zeroes of *f*(*z*). If coincides with *D*,
our theorem has been proved. However, if comprises only a part
of the domain *D*, then there exists a boundary point *b*
of the set , which is an internal point of *D*. There
exists a sequence of points *b*_{n} of
the set which converges to *b*; by the continuity of *f*(*z*),
the point *b* is a zero of *f*(*z*). On the
other hand, *f*(*z*) is not identically equal to
zero in some neighbourhood of the point *b*, because then *b*
would be an internal and not a boundary point of . Hence,
it follows by Theorem 1 from this that in some neighbourhood of
the point *b *is not a zero of *f*(*z*),
which contradicts the fact that *b* is a boundary point of
.
This contradiction also proves the uniqueness theorem.

It follows from the uniqueness theorem that a
function *f*(*z*), which is analytic in some domain
and does not vanish identically, cannot vanish in some sub-region
of *D* nor on some arc, lying in *D*, not even on a
sequence of points of *D*, converging to its internal
point.

However, it is easy to produce an example when an
infinite sequence of zeroes of a function converges to a boundary
point of its domain of analyticity: The function *f*(*z*)
= sin 1/*z *becomes zero on the sequence of points *z*
= (1/*n**p*) (*n*
= ±1, ±2, ···), which converges to the point *z* = 0.