All basic concepts of the theory of functions of a complex
variable are introduced: The concept of function, its derivative,
integral, etc. The reader will see that the basic definitions,
known from the analysis of the functions of a real variable
remain almost without change, but their contents change in a very
essential way. Thus, there remains the ordinary, geometrical
illustration of functions by curves in the plane and the concept
of functions as mappings of plane sets (Chapter IV). The
condition of the differentiability of functions of a complex
variable turns out to become significantly more rigid than that
of functions of a real variable (Chapter V). For example, there
follows automatically from the conditions of differentiability in
the complex domain the existence of derivatives of all orders
(Chapter XVII) and all the properties of functions, which are
quite unusual for real analysis (**14.**, **15.**,
etc.).

Complex numbers and functions of a complex variable already occur in studies as early as the XVIII-th Century. Especially, Leonhard Euler used them in his numerous studies, which rightly must be classed as theories of function of a complex variable. His remarkable publications studied in detail the elementary functions of a complex variable, including the logarithm, exponential, trigonometric functions and their inverses (1740 - 1749), presented the conditions of differentiability (1755) and the beginning of the integral calculus of these functions (1777). The conditions of differentiability were already referred to in1752 by Jean le Ronde d'Alembert in connection with hydrodynamic work. However, Euler was the first to explain the general character of the differentiability conditions. Moreover, he presented many applications of the theory of functions of a complex variable to different mathematical problems, proposed first their application to hydrodynamics (1755 - 1757) and cartography (1777).

After Euler had presented his results and methods, improved and systematized them, also at the start of the first half of the XIX-th Century, the theory of the functions of a complex variable became an important ingredient of mathematical analysis. The basic contributions belong to Cauchy and Karl Weierstrass (1815 - 1897) who developed the integral calculus and the representation of functions by series. Likewise, one must mention here Bernhard Riemann who dealt with the geometric questions of the theory of functions and their applications.

**1. Complex Numbers**** **For the reader's
convenience, we present here the basic definitions and facts
relating to the concept of complex numbers, their operations and
geometrical illustration.

The first understanding of
"imaginary" numbers as well as of square roots of
quadratic negative numbers occurred in 1545, still in the XVI-th
Century (Facio Cardano 1444 - 1524). By the middle of the
XVIII-th Century, complex numbers were only used incidentally in
the work of different mathematicians (I. Newton, N. Bernoulli, A.
C. Clairault). The first account of the theory of complex numbers
in Russian belongs to L. Euler ("Algebra", St.
Petrograd 1763, later editions in other languages ): The symbol
"*i*" was also introduced by Euler. The
geometrical interpretation of complex numbers occurred in 1799
(Kaspar Wessel, 1745 - 1818).

**1.1
Complex Numbers**** **One
calls *complex numbers *expressions of the form *x*
+ *iy*, where *x *and *y,*. the *real *and
*imaginary *parts of the complex number *x* + *iy,
*are denoted by

In particular, if *y* = 0,
then* x* + *i*0 is assumed to coincide with the
real number *x*; if *x = *0, then 0 + *iy*
is simply denoted by *iy *and is said to be *purely
imaginary*.

We will now define for the set of
complex numbers the concept of equality and the simplest
operations. We will say that the complex numbers *x*_{1}
+ *iy*_{1}* *and* x*_{2} + *iy*_{2}*
*are *equal*,

if and only if *x*_{1}
*= x*_{2}* *and* y*_{1}*
= y*_{2}*.*

However, note that if* x*_{2}
*= x*_{1}, but *y*_{2}* = -y*_{1}*,
*the complex number *x*_{2}+ *iy*_{2}
is said to be *conjugate *to *x*_{1} + *iy*_{1}*
*and denoted by Thus

We now continue with the operations on complex numbers.

1) **Addition: ***Sum **z*_{1} + *z*_{2}* *of
the complex numbers *z*_{1} = *x*_{1}
+ *iy*_{1}* *and* z*_{2} = *x*_{2}
+ *iy*_{2}* *is called the complex number

The stated definition yields at once the laws of addition:

a) *Commutative Law*: *z*_{1}
+ *z*_{2}* *= *z*_{2} + *z*_{1}*.
*b)

Addition admits the inverse operation: One may
likewise find for any two complex numbers *z*_{1}
= *x*_{1} + *iy*_{1}* *,*
z*_{2} = *x*_{2} + *iy*_{2}*
*a* *number *z *so that *z*_{2}
+ *z *= z_{1}. This number is called the *difference*
of the numbers *z*_{1} and *z*_{2}
and denoted by *z*_{1} - *z*_{2}*.
*Obviously,

2) **Multiplication: **P*roduct z*_{1}*z*_{2
}of the complex numbers *z*_{1}
= *x*_{1} + *iy*_{1}* *and*
z*_{2} = *x*_{2} + *iy*_{2 }is
called the complex number

This definition yields the laws of multiplication:

a) *Commutative Law*: *z*_{1}*z*_{2}*
*= *z*_{2}*z*_{1}*.
*b)

If *z*_{1}* *and *z*_{2
}are real numbers, Definition (6) coincides with the
ordinary definition. For *z*_{1} = *z*_{2}*
= i*, (6) yields

It is readily observed that (6) is obtained by rearrangement
following the ordinary rules of algebra and replacing *i*·*i
*by -1. Note yet that the product of a complex number *z*
= *x *+ *iy *by its conjugate is always negative.
In fact, by (6):

Multiplication yields likewise the inverse operation,
provided, however, that the given factor is not zero. Let *z*_{2 }¹
0; then one may find the number *z *such that *z*_{2} *z*_{
}*= z*_{1}; by (6), one must solve for this
the system of equations

which for *z*_{2 }¹ 0 is always uniquely soluble, since its
determinant *x*_{2}²* *+ *y*_{2}²
> 0. This number *z* is called the *quotient *of
the two numbers *z*_{1}*, z*_{2}
and is denoted by *z*_{1}/*z*_{2}.
Solving System (9), we find

It is readily seen that (10) may be obtained by multiplying
the numerator and denominator of the fraction *z*_{1}*/z*_{2}
by* *

3) **Integral** **Power **The
product of *n* equal numbers *z* is called the *n*-th
*power *of the number *z *and denoted by *z*^{n}:

The inverse operation -* taking the root* - is defined
as follows: The number *w* is called the* n*-th
root of *z*, if *w*^{n} = *z *(
it is denoted by , where for *n* = 2 one uses simply ). We will
see below that the root has for every *z *¹ 0 *n* different values.

We can now rewrite Equation (7) in the form *i*² = -1
and obtain for *i*

(here denotes one of its two possible values).

**2
Geometrical illustration****
**Consider the plane of Cartesian
co-ordinates *xOy *and represent the complex number *z*
= *x* + *iy *by the point with the co-ordinates (*x,
y*). For this, the real number will be represented by a point
of the *x*-axis (which we will call in the sequel the *real axis*), the imaginary number by a point of the *y-*axis
(called the *imaginary
axis*). In particular, the image of
the number *i* will be the point (0, 1) on the imaginary
axis.

It is readily seen that also, conversely, each
point of the *xOy *plane with co-ordinates (*x*, *y*)
will thus be made to correspond to a definite complex number *z*
= *x* + *iy *so that the correspondence between the
set of all complex numbers and all points of the plane is
mutually single-valued. Hence, in the sequel, we will not
distinguish between the concept of complex numbers and the points
of the plane and will speak, for example, of "the point 1 + *i*
", "the triangle with vertices *z*_{1}, *z*_{2},
*z*_{3}", etc.

Moreover, every point (*x*, *y*)
corresponds to a completely defined vector - the radius vector of
this point - and corresponding to the radius vector, lying in the
plane, there corresponds a definite point - its end point (Fig.
1). Hence, in the sequel, we will represent complex numbers
likewise in the form of radius vectors in the plane.

Fig. 1 explains the geometrical meaning of the
the operations of addition and subtraction of complex numbers:
The sum and difference of the complex numbers *z*_{1}*
*and* z*_{2} are represented, respectively, by
the vectors in the directions of the diagonal of the
parallelogram, formed by the vectors* z*_{1} and *z*_{2}.

In the sequel, side by side with the
representation of complex numbers in Cartesian co-ordinates, it
will be useful to employ their representation in polar
co-ordinates. Obviously, we combine for this the the polar axis
with the positive *x*-axis, the pole with the origin of
the co-ordinates; then, if we denote by *r* the polar
radius and by *j* the polar angle through the point z (Fig. 1), we have

The polar radius __r__ is called the *modulus**
*of the complex number* z *and denoted by |*z*|,
the angle *j* its *argument** *and denoted by Arg *z*. While the
modulus of a complex number is uniquely defined by

its argument is only determined apart from an arbitrary
multiple of 2*p *:

arctan denotes here the principal value Arctan, i.e., the
value larger than *-**p*/2
and not exceeding *p*/2, *k*
is an arbitrary integer. In the sequel, apart from the symbol
Arg, we will denote thus all values of the argument (cf. **6.**).

The following inequality is obvious (cf. Fig. 1):

The equality sign in (4) applies if and only if Arg *z*_{1}
= Arg *z*_{2}.

It follows from (6) of **1. **that
during multiplication of complex numbers their moduli are
multiplied and their arguments are added. In fact, one has:

Hence, during multiplication of the complex number *z*_{1}
by *z*_{2}, the vector *z*_{1} is
stretched |*z*_{2}| times and, in addition, it is
rotated counter-clockwise by the angle arg *z*_{2}.
In particular, multiplication of the number *z* by *i*
reduces to a rotation (without stretching) of the vector *z*
by a right angle counter-clockwise.

Fig. 2 shows the construction of the product *z *= *z*_{1}*z*_{2};
in order to obtain *z*, it is sufficient to construct with
the segment *Oz*_{1 }as base the triangle *Oz*_{1}*z*,
similar to the triangle *O*1*z*_{2}.

If |*z*_{2}| < 1, then *z*_{1}shrinks
effectively 1/|*z*_{2}| times.

Moreover, division of a complex number *z*_{1 }by
*z*_{2} reduces to multiplication of *z*_{1}by
1/*z*_{2}, whence one may limit the explanation of
the geometrical meaning of the operation *w* = 1/*z*.

To start with, let |*z*| < 1 (Fig. 3). We draw from
the point the perpendicular to *Oz* and* *through
the point *z* of intersection
of the perpendicular with the circle |*z*| = 1 the tangent
to this circle. Obviously, we have for the point of intersection *w *of this tangent with the ray *Oz*

and deduce from the similarity of the triangles *Oz**z *and *Oz**w *that ||*w*
/*z*|| = ||*z*
/ *z*||, whence

|*w*| = 1/|*z*|,

because |*z*| =
1. Thus, the number *w *is
conjugate to 1/*z ,**w*
= 1/
and, in order to obtain the point *w* = 1/*z, *one
must construct the point symmetric to *w *with
respect to to the real axis.

The process of going from the point *z* to the point *w = *1/ is called *inversion** *or *symmetry
with respect to the unit circle *|*z*|* *= 1*.
*Thus, the operation *w* = 1/*z *reduces
geometrically to the execution of two consecutive symmetries -
inversion and symmetry with respect to the real axis.

If |*z*| > 1, the construction must be executed in
the reverse order; if |*z*| = 1, the point *w* = 1/ coincides with *z*
and the construction of *w* = 1/*z *is reduced to
symmetry with respect to the real axis.

The geometrical meaning of raising to a power
follows from the preceding results. We note regarding the
construction of the *n*-th root of *z *that we have
from the definition of the root and (5) for *w** *= (*z*)^{1/n}

The first of these relations shows that the modulus of all
roots have the same sign, the second that their arguments differ
by a multiple of 2*p*/*n*,
because one may add to the value of arg *z* a multiple of
2*p**,* whence the *n*-th
root of any complex number *z* ¹
0 has *n* different values and these values lie at the
vertices of the triangles inscribed in the circle |*w*| =
|*z*|^{1/n}*.*( Fig. 4 where *n
*= 6)

**1.2 Functions of a complex variable**** **We will now
introduce more fundamental concepts of the theory of functions of
a complex variable: The concept of function of a complex
variable, its limit, derivative and, finally, the concept of an
analytic function. The centre of these concepts is the theorem
which establishes the conditions of the differentiability of
functions of a complex variable, usually referred to as the
Cauchy-Riemann. conditions which, however, were employed*
prior *to Cauchy and Riemann in a very essential manner by
d'Alembert and Euler, whence we will call them the *conditions
of d'Alembert and Euler*, following
a suggestion by A.I.Markushevich. .

**3.
Geometrical concepts **One calls
*region* in the complex plane the set *D *of the points
with the properties: 1) there belongs to every point of *D*
of this set also a sufficiently small circle with centre at this
point (*property of **neighbourhood*), 2)
any two points of D may be joined by a path consisting of point
of D (*property **connectivity*).

As simple examples of regions may serve
neighbourhoods of points in the complex plane. We understand by
the *e* -* neighbourhood*
of the point *a* an open circle of radius *e** *with centre at the
point, i.e., the set of points *z* which satisfy the
inequality

We call *boundary points**
*of the region *D *those points which themselves do
not belong to *D*, but in any neighbourhood of which lie
points of this region. The set of boundary points of *D*
is called the *boundary *of this region. The region *D *together
with its boundary is denoted by and called the *closed region*.

We will assume that the boundary of a region consists of a
finite number of closed lines, sections and points ( we do not
present definitions of these concepts, cf. Fig. 5, where the
boundary of the region consists of three closed lines G_{0}, G_{1},
G_{2}, two cuts g_{1}, g_{2}
and a single point *a*). The
lines and cuts, comprising the boundary, will always be assumed
to be *piecewise smooth*,
i.e., to consist of a finite number of smooth arcs (with
continuously changing tangent). In the case of the bounded region
*D*, the number of connected parts, into which its
boundary is divided, is called the *order of connectivity *of
the region (Fig. 5 shows a 5-connected region; *G*_{0 }and *g*_{1} form one connected
part of the boundary). In particular, if the boundary of the
region *D* is connected (i.e., consists of a single
connected part), *D *is said to be a *simply connected domain*.

Regarding the definition of the order of
connectivity of unbounded regions, cf. **24.
**(A region *D* is said
to be *bounded*, if it belongs to some circle |*z*| < *R.*)

Let *D *be a* simply connected region *and
*G *its boundary. Select on *G* a point and, starting from this
point, move around *G** *in the positive
direction.Assume that the *positive direction**
of the circuit around the boundary of the region* is that for
which the region lies all the time on the left hand side. Then,
for certain points, *G *will only be travelled along once (for example, *A *in
Fig. 6), other several times (for example, *B *twice, C
three times). We will call points of the first kind *simple*,
of the second kind *multiple points* of the contour *G*, where
the number of times the point passes are called its *multiplicity*
(*B* is a double, C a triple point), The concept of
multiplicity of boundary points also extends to simply-connected
regions.

**4. Functions of a complex variable**** **One says that there is given on
the set *M *of points of the *z*-plane the function

if there is stated the law by which to every point *z*
of* M *there corresponds a point or a set of points *w*.
In the first case, the function *w* = *f*(*z*)
is said to be *single-valued*, in the second case *multi-valued*.
The set *M *is called the *set of definitions *of
the function *f*(*z*) and the union *N *of
all values *w*, which *f*(*z*) assumes on *M*
the *set of its changes*. In the sequel, a more important
role will have the case when the sets* M *and *N*
are domains (cf. Theorem in **5.**)

If we set *z* = *x* + *iy*, *w* = *u*
+ *iv*, the definition the function of the complex
variable w = f(z) will be equivalent to giving two functions of
two variables

We agree to allot the values *z* to one, those of *w*
to another complex plane. Then, one may represent a function of a
complex variable geometrically by some *mapping**
*of the set *M* of the *z*-plane onto a set *N*
of the *w*-plane. If the function *w* = *f*(*z*)
is single-valued on the set *M *and for this two different
points of *M* always correspond to two different points of
*N*, such a mapping is *mutually
single-valued or single-sheeted** *in *M.*

Let there be given the function *w* = *f*(*z*),
which maps the set *M *onto the set *N*, and *w *= *g*(*w*) the set *N*
on *P. *The function *w = **h*(*z
= g*[*f*(*z*)], mapping *M* onto *P
is *called* the **compound
function*, consisting of *f* and *g, *and
the corresponding mapping* h *the* **superposition of the
mappings** f and g* (Fig. 7). In particular, if
the mapping *w* = *f*(*z*) is mutually
single-valued, the functions *z* = *j*(*w*) - the inverse of *f* - then

**Example **The *linear function** *is defined throughout the *z*-plane
by the relation

where *a* and *b *are arbitrary complex
constants. Set *k* = |*a*|, *a*
= Arg *a*, i.e., *a* = *k*(cos *a** + i*sin *a*) and represent Function (5) by
the compound function, involving the functions

a) *z*_{1}* = *(cos *a* + *i *sin *a*)*z*,

b) *z*_{2}* =* *kz*_{1}*,
*c

Recalling the geometrical meaning of
multiplication (**2**),
we see that the mappings *a*) and *b*) cause rotation of the *z*-plane by
the angle *a *and a similarity
mapping of the *z*_{1}*-*plane with the
similarity coefficient *k*. The mapping c) is
geometrically a displacement of the entire *z*_{2}*
-*plane by the constant vector *b.*

The linear mapping (5) is itself a superimposition of the described mappings (Fig. 8), whence Mapping (5) is mutually single-valued in the entire plane and transforms straight lines into straight lines (where the angle between two lines is preserved) and circles into circles.

**5. Differentiability and analyticity**** **Let the function
*w* = *f*(*z*) be defined and single-valued
in some neighbourhood of the point *z*_{0}*
= x*_{0}* + i* *y*_{0}* *except,
may be, at the point *z*_{0}*.*

We will say that the exists the *limit of the function**
f*(*z*) *as z *® *z*_{0}*
*( notation: ) if there exist the limits

we will write this

Since our definition reduces to the ordinary definition of a limit of a real function, the basic properties of the limit formation is retained for functions of a complex variable. In particular, we have

A definition of limit may be likewise formulated with the aid
of the concept of neighbourhood: if and only if for any
*e *> 0 there is a *d **> *0* *such
that for all points in the *d*-neighbourhood
of *z*_{0 }(except, may be, *z*_{0 }itself)
the corresponding point *w*_{0 }lies in an *e*-neighbourhood of *w*_{0};
in other words, if there follows from the inequality

that

We emphasize that by our definition the function *f*(*z*)
tends to its limit* independently of the way* in which the
point z approaches the point *z*_{0}. In other
words, if the limit exists, then as *z *tends to *z*_{0
}according to any law (for example, along any curve or by
any sequence), *f*(*z*) will have the same limit.

A function *f*(*z *)is said to be *continuous
at the point z*_{0 }(including the point *z*_{0
}itself) if it is defined in some neighbourhood of *z*_{0
}(in cluding the point *z*_{0 }itself) and

Obviously, it is necessary and sufficient for the
continuity of *f*(*z*) at the point *z*_{0
}that the functions *u*(*x, y*) and *v*(*x,
y*) are continuous at the point (*x*_{0}, *y*_{0}).

The function *f*(*z*) is said to be
*continuous in the region D* if it is continuous at every
point of the region. We note without proof that for functions,
continuous in closed domains, and likewise on closed lines or on
segments of a line, including its end points, there hold the
ordinary properties of functions, continuous on closed intervals.
Namely, every function *w *= *f*(*z*),
continuous on a closed set ,

1) is* bounded on it*, i.e., there exists
a constant *M* such that for all *z* on

2)* its modulus attains its largest or smallest value*,
i.e., there exist in points *z*' and *z" *such
that for all *z* in

3) is *uniformly continuous*, i.e., for an
arbitrary *e* > 0, there
exists a number *d *> 0,
which depends only on *e,* such
that for any pair of points *z*_{1} and *z*_{2}*
*of , satisfying the inequality |*z*_{1} -
*z*_{2}| < *d,*

We note likewise without proof one proposition which will be used frequently in the sequel *.

* Its proof is obtained by topological methods.

4) **Theorem:** *If the
function w = f*(*z*) *is continuous in the region D
and provides a mutually single-valued mapping of this region onto
some set **D** in the w-plane, then **D **is likewise a
domain and the inverse function z = **j*(*w*) *is
continuous in **D.*

Let the function *f*(*z*) be
defined in some neighbourhood of the point *z*. We will
say that *f*(*z*) is *differentiable* at the
point *z* if there exists the limit

This limit will be called the *derivative*
of the function *f*(*z*) at the
point *z*.

The condition of differentiability of the
function *f*(*z*) at the point *z *in terms
of the real functions *u*(*x, y*) and *v*(*x,
y*) is expressed by the

**Theorem: ***Let
the function f*(*z*) = *u*(*x, y*) + *iv*(*x,
y*)* be defined in some neighbourhood of the point z,
where the functions u*(*x, y*) *and* *v*(*x,
y*) *are differentiable. Then, it is necessary and
sufficient for the differentiability of the function of the
complex variable f(z) at the point z that at this point apply the
relations*

(the *d'Alembert
- Euler relations*).

Equations (10) were obtained in connection with hydrodynamic problems by d'Alembert (1752) and Euler (1750; in 1777, Euler again obtained these equations while studying the integrals of functions of a complex variable.

**a)** **Necessity: **Let
there exist

We employ the remark relating to the independence
of the limit on the methods of approximating it. Firstly, we
assume that the point *z + h *approaches the point *z*
along a straight line, parallel to the real axis, i.e., as *h*
= *s* ® 0, staying real. We
then obtain:

We will now find the same limit under the assumption that the
point *z + h *approaches *z *along a straight line,
parallel to the imaginary axis, i.e., that *h* = *it *and
*t *® 0, while remaining real.
We find:

Comparing Expressions (11) and (12) for *f *'(*z*),
we obtain:

whence follows (10) (cf. **1.**
relating to the definition of equality of complex numbers).

**b) Sufficiency: **By the definition of the
differential of a functions of two real variables, we have:

where *a* and *b *tend to zero together with *h
= s + it*. Then the increment of the function *f *(*z*)
becomes:

where *h *= *a **+
i**b*. Using (10), this
increment may be rewritten in the form

where is a completely defined number, which does not
depend on *h,* and *h *tends
to 0 together with *h*. Dividing (14) by *h*,* *we
see that exists and equals *A*. Thus, the theorem has
been proved.

Using the d'Alembert-Euler condition, the derivative of *f *(*z*)
may be represented in the forms:

Since the usual properties of the algebraic operations and the limiting process are maintained during the transition to functions of a complex variable, also the ordinary rule of differentiation is preserved, the derivation of which is only based on the properties above:

(in the last formula, *f* and *j
*denote mutually inverse functions , where it is
assumed that they realize single-sheet mappings concerning
neighbourhoods of the points *z *and *w*).

A function *f*(*z*), which is differentiable at
every point of some domain *D*, is said to be *analytic *(or *regular*)
in this domain. We emphasize that our definition of analytic
functions assumes that it is single-valued in *D*, because
the concepts of limit and derivative are only defined for such
functions.

In **25.**,
we will generalize the concept of analyticity by extending it
also to multi-valued functions, but apart from there we will
always assume functions to be single-valued.

We note in conclusion one generalization of the
d'Alembert-Euler condition. Let there be given a function *f*(*z*)
which is differentiable at the point *z*; we assume
arbitrary directions, characterized by unit vectors *s*^{0}*
*and* n*^{0} (i.e., complex numbers with
moduli 1) and such that rotation of *s*^{0}* *and*
n*^{0} is performed by a right counter-clockwise
angle (i.e., *n*^{0}* = is*^{0}).
Using the fact that the calculation of the derivative does not
depend on the direction, we obtain, taking the derivative once in
the direction *s*^{0}* , *another time in
the direction *n*^{0}):

(
being derivatives of function of two real variables in the
corresponding directions); the derivation of (17) is analogous to
that of (11) and (12). Setting *n*^{0}* = is*^{0}
and comparing in (17) real and imaginary parts, we obtain

These equations are generalizations of the
d"Alembert-Euler conditions which we wanted to note. In
particular, setting in them *s*^{0}* = 1, n*^{0}
= *i*, we obtain (9).

We will still present the d'Alembert-Euler conditions in polar
co-ordinates (*r*, *j*).
Let *s*^{0}* *be the unit vector of the
tangent to the circle |*z*| = *r* in the
counter-clockwise direction and *n*^{0}* *the
inward normal vector to the circle; then

and (18) assumes the form

**1.3
Elementary Functions**** **This section treats the elementary functions of a
complex variable and their geometric illustrations - by mappings.
These functions are natural extensions into the complex domain of
the ordinary, elementary functions of analysis. However, thus
extended functions have sometimes new properties, for example,
the exponential function of a complex variable *e*^{z}*
*turns out to be periodic, the functions sin *z* and
cos *z* cease to be bounded, the logarithm becomes
negative numbers (and, in general, complex numbers other than
zero) etc.

Special interest is attached to
the study in the complex domain of multi-valued functions,
because only such a study permits to explain their
multi-valuedness. We will limit consideration here to different
examples of multi-valued functions and by means of these examples
demonstrate the possibility of isolating single-valued branches,
which turn out to be analytic functions. Only in **25.****, **we
will see the general concept of multi-valued analytic functions
and then may consider not only branches, but these functions
themselves as analytic functions.

On the whole, the theory of the elementary functions of a complex variable was established by Leonhard Euler in his work during the Forties of the XVIII-th Century. We must note that this work of Euler was carried out ahead of our time; for example, his theory of the logarithm only became known with great effort only quite recently.

In **7.**, we
specially treat the simple fractional rational function *w*
= ½(*z* + 1/*z*) in view of its important role in
practical problems *cf. the following Chapters II and III). A
very successful application of this function is connected with
the work of Nikolai Egonovich Joukowski (1847 - 1921), whence we
will call it the *Joukovsky function.*

**6. The Functions ****w****
= ****z**^{n}
**and** **w = z**^{1/n}**,**** **where *n* is any
positive integer, has already been defined in **1.**** **for all complex *z*. The
first of these functions

is single-valued. If we introduce in the *z-* and *w-*planes
polar co-ordinates, setting *z* = *r*(cos*j *+ *i*sin*j*), *w* =
*r*(cos*q *+ *i
*sin*q*), Relations (1) may
be rewritten in the form:

which link real quantities.

It is seen from (2) that the mapping, generated by the
function *w *= *z*^{n}*, *reduces
to a rotation of every vector *z* ¹
0 by the angle (*n* - 1) arg *x *and its extension
by |*z*|^{n-1}.Moreover, it is obvious
that points *z*_{1}* *and *z*_{2}*
*with different moduli and arguments, differing by an
integral multiple 2*p*/*n*,
and only such points, become for Mapping (1) a single point.
Consequently, it is necessary and sufficient for a single sheet
mapping *w* = *z*^{n}* *in
some region *D* that *D *does not contain either of
the two points *z*_{1}* *and *z*_{2}*,
*linked by

This condition is satisfied, for example, by the sectors

each of which* *is transformed by the mapping *w *=
*z*^{n} in the *w*-plane excluding
the positive semi-axis. Then, all rays with vertex at *z*
= 0 become rays with vertex at* w* = 0 (only are turned by
a certain angle), and all arcs of circles with origin at *z*
= 0 become arcs of circles with centre *w = *0 (only,
generally speaking, with another radius). Fig. 9a shows the
prototype in one such sector of the *z*-plane of the set
of polar co-ordinates in the *w*-plane.

The formula

equivalent to (1) yields for the straight lines *u* = *u*_{0}*
*and *v* = *v*_{0} in the *z*
-plane the corresponding curves as polar equations

They are shown in Fig. 9 b) (the first - dotted, the second -
dashed lines); for *n* = 2, they are ordinary hyperbolae.

Finally, we note that the function *w* = *z*^{n}
is analytic in the entire plane, because there exists for any *z*

The function

the inverse of the function *z* = *w*^{n}*
,* is *n*-valued for *z* ¹
0. As follows from **2****.**,**
**the value of the root is determined by the
value of the argument, taken for the point *z*. We denote
here by arg *z*_{0 }one of the values of the
argument at the point *z* ¹ 0
and assume that the point *z* describes, for example, with
*z*_{0} some continuous curve *C*, which
does not pass through the origin. We will denote by arg *z*
that value of it, which changes for this continuously, starting
with the value arg *z*_{0}*. Due to the continuity
of arg *z* and |*z*|, the value *w* = , which
is completely determined by the choice of the argument made,
likewise changes continuously.

*Obviously, this value for fixed *z*_{0},
C and arg *z *is uniquely determined.

We assume that the curve *C* is closed
and does not contain inside it the point *z* = 0. Then,
during a complete circuit of the point *z*, the point *w*
= ,
where
is chosen according to our knowledge of the root, describes some
closed curve *G*, which reverts
to its initial point, because for this arg *z* reverts to
the initial value arg *z*_{0}. Knowledge of the
root, determined by by another choice of the initial value arg *z*_{0}
(differing from the preceding one by an integral multiple of 2*p*) for a complete circuit *C, *obviously,
also describes closed curves *G*_{k},
differing from *G *only by a
rotation by an angle 2*k**p*/*n*.
*k *= 1,2, ··· , *n*-1 (solid curves in Fig. 10
below).

Let
be a closed curve without intersections, containing *z* =
0 inside, and *z*_{0} some point on the curve . Then, for
a complete circuit of , starting at *z*_{0} in
the positive direction,. the corresponding point *w* = (the
value of the root is determined just as above) does not return to
its initial position and assumes a new value *w*_{0}^{1},
where

*w*_{0}^{(1) }=(cos2*p*/*n+i*sin 2*p*/*n*) *w*_{0}^{
}- the value of other than *w*_{0}.

This is explained by the fact that arg *z *undergoes
during a circuit of the increase 2*p*.
The point *w* = reverts to its iinitial position after
the *n*-th circuit of the curve ( in Fig. 10 below the
broken line).

Hence, in any region *D*, which does not
contain a closed curve around the point *z* = 0, one may
select *n* *continuous and single-valued functions*,
which assume one of the values . These *n*
values are called the *branches *of
the multi-valued function *w* = ; their values at every
fixed point differ from each other by the factor (cos 2*k**p*/*n + i*sin 2*k**p*/*n*)*. *Obviously,
each such branch will realize a single sheeted mapping of the
region *D*, whence there is applicable at each point of
this region the theorem of the derivative of the inverse function
(cf. **5.**)
, according to which there exists a completely definite value of
the derivative

or, if it is agreed to write

Thus, any of the constructed branches is an
analytic function in the region *D*.

In the region D of the type under consideration,
also the infinitely valued function Arg *z*
becomes an infinite set of continuous and single-valued branches.
We will denote each such branch by the symbol arg *z* and
use it each time as this branch is chosen.

However, if the region *D* contains just
one closed curve, surrounding the point *z* = 0, that in
this region the branches of the function cannot be separated
from each other. In fact, if in the neighbourhood of some point *z
*# 0 of *D* we also select
some branch (for a sufficiently small neighbourhood of the point *z
* 0 this can be done), then moving along the
curve, approaching *z* = 0, we come to another branch.
Consequently, in such a region *D*, we may not consider in
a similar manner to the preceding case the function as a set
of separate (single-valued) analytic functions. Then, the point *z*
= 0, in any neighbourhood of which it is impossible to separate *n
*different branches of the function (the branches would
join at this point), is called the *branch*
*point** *of this function.

As an example of a region *D* of the first
type, we may consider the *z*-plane with a cut along the
line *L*, running from *z* = 0 to infinity. If *L
*coincides with the positive semi-axis, then these branches *w*
=
map the region *D *onto the sectors

These mappings are the inverse to those
considered above with the aid of the function *w = z*^{n}.

The region *D* is deliberately a region of
the second type, if it contains the point *z* = 0 inside
it.

**7.
Joukovsky's function ****w = ****½(****z
+ 1/z****)** is defined and
single-valued for all *z *0; obviously, it is analytic for
those *z*. We will find the regions of the single
sheetedness of the mapping

We assume for this purpose that *z*_{1} and *z*_{2}*
*become the single point *w*; we will then have *z*_{1
}+ 1/ *z*_{1}= *z*_{2 }+ 1/ *z*_{2},
whence

and, consequently,

Thus, it is necessary and sufficient for single-sheetedness if
the mapping (1) at any point that *D* does not contain any
two points *z*_{1} and *z*_{2},
linked by the relation *z*_{1}*z*_{2}
= 1.

For example, this condition is met by the inside of then unit
circle |*z*| > 1. In order to study Mapping (1), we set
*z* = *r*(cos *j*
+ *i*sin *j*), *w*
= *u* + *iv *and separate the real and imaginary
parts. The mapping (1) then becomes

and we see that every circle |*z*| = *r*_{0}
< 1 becomes the curve

i.e., an ellipse with the semi-axes *a* =
½(*r*_{0} + 1/*r*_{0}),b = ½(1/*r*_{0}
- *r*_{0}), passed in a negative direction*. As *r*_{0
}1, this ellipse
contracts into the segment [-1, 1] of the *u*-axis, for *r*_{0
}0, it moves to
infinity. Consequently, the function (1) maps the inside of the
circle |*z*| < 1 onto the outside of the segment [-1.
+1] (Fig. 11). All internal points of this segment are double
points (cf. **3.**)
and this may be assumed to constitute the two shores: The
function (1) maps the upper half of the circle |*z*| = 1
onto the lower shore, the lower half onto the upper shore.

* This sis the meaning of the (-) sign in the second equation (4).

We note still that the radii arg *z* = *j*_{0}*
*, 0 < *r* < 1 becomes by the mapping (1)
branches of the hyperbolae

(Fig. 11). The foci of these hyperbolae, just as those of the ellipses (4), lie at the ends of the segments [-1, +1].

It also follows from (3) that the circles |*z*|
= *r*_{0 }> 1 map into the ellipses with
semi-axes *a = *½(*r*_{0} + 1/*r*_{0}),b
= ½(1/*r*_{0} - *r*_{0}). These
ellipses coincide with those into which map the circles |*z*|
= *r*_{0 }< 1, except that they are travelled
along in the positive direction. Consequently, the function (1)
maps also the outside of the circle *z* > 1onto the
outside of the segment [-1, +1] of the *u*-axis,
where the upper semi-circle becomes the upper shore , the lower
one the lower shore.

Returning to (1), we see that the function

is double-valued - to every point *w*,
there belong two points *z*_{1} and *z*_{2}
,* *linked by the condition *z*_{1} *z*_{2}*
= *1 (cf. (2)). This double-valuedness is brought about by
the presence in (6) of the square-root. If we set the other value *z*
corresponding to *w* will be and it is immediately seen that *z*_{1}*z*_{2}*
= *1*.*

Denote by *r *_{1}*,q*_{1
}and* r*_{2}*,
q*_{2}, respectively, the moduli and arguments
of the complex numbers *w* - 1 and *w* + 1 (Fig.
12). Then, the moduli and arguments of the root in (6) will be and (*q*_{1} + *q*_{2})/2
(cf. (**2.**)
regarding the rule for the extraction of the root). Hence, for
the passage of the point *w* along a closed curve of type
I and II (Fig. 12), encircling only one of the points +1 and -1,
the sign of the value of the root changes to the opposite sign.
In fact, for such a circuit, *q*_{1}
(or *q*_{2})
changes by 2*p, *but *q*_{2} (or *q*_{1}) do not change,
whence the argument of the roots changes by *p*,
the modulus of the same root for the passage along any closed
contour reverts to its initial value.

Now, if the point *w* moves along a curve of Type III
(Fig. 12), surrounding both points ±1, the values of the roots
do not change, because then *q*_{1}
and* q*_{2}
change by 2*p* and,
consequently, the argument of the root *(q*_{1}
+ *q*_{2})/2
likewise changes by 2*p. *The
value of the root does not change in the case when z travels
along a curve of Type IV (Fig. 12) and does not contain inside it
either of the points ± 1, because neither *q*_{1}
nor* q*_{2}
change.

Thus, in any domain *D*, in
which it is impossible to produce a closed curve, surrounding
only one of the points +1 or -1, the function (6) admits division
into two single-values *branches*. At every fixed point *w*,
these branches differ in the sign of the root in (6) and yield
two values *z*, linked by the condition *z*_{1}*z*_{2}*
*= 1. Each of these two branches yields a single-sheeted
mapping and, by the theorem concerning the derivative of the
inverse, are analytic functions.

However, if one may move in the region *D*
around the point +1 ( without moving around the point -1) or -1
(without moving around +1), for example, if *D
*contains either one of these points, then in such a
region branches of the function (6) cannot be separated from each
other. The points *w = *±1, at which both branches of the
function (6) would join, is called the *branch point *of
this function.

As an example, we consider a region *D*
of the first type with the cut *L*,
joining the points -1 and +1. If *L *is
the segment [-1, 1] of the real axis, then the two branches of
(6) map *D* onto the inside and
outside of the unit circle, respectively. These mappings are
inverse in the sense considered above (Fig. 11).

**8.
Exponential function and logarithm**** **We
define* the *exponential function e*^{z }for
any complex number *z* = *x* + *iy *by the
relation

* We recommend the reader to whom our definition of the exponential function is too formal, to define it in analogy with the definition of that function of a real variable

Then, one must
prove the existence of the limit of the sequence of the complex
variables *z*_{n} = (1 + *z*/*n*)^{n}
for any* z *and compute this limit. This is most simply
done as follows: By the power rule, we have

and arg *z*_{n}*
*= *n *arctan (*y*/*n*)/(1 + *x*/*n*);
converting in the first expression the small of higher order term
(*x*²* *+ *y*²)/*n*² into powers
and replacing in the second small angle by its tangent (*y*/*n*)/(1
+ *x*/*n*), we see that there exist the limits

However, from the
existence of these limits follows the existence of the limit (*),
whence we find that |*e*^{z}| = *e*^{x
}and arg *e*^{z }= *y*;
this coincides with the formula (1).

We will now show that: 1) for real *z = x *our
definition coincides with the ordinary one; 2) our defined
function is everywhere analytic; 3) there is retained the
ordinary differentiation rule

4) the basic property of the exponential function (*addition
theorem*)

is preserved.

The first property follows directly from (1), if
one sets there *y* = 0, the second from the theorem in **5.**
, because there hold at any point in the plane the
d'Alembert-Euler conditions

For the proof of 3), using the independence of
the derivative from the direction, we compute *e*^{z}
in the direction of the *x*-axis:

which also yields (2).

Finally, we set for the proof of 4) *z*_{1}
= *x*_{1} +* iy*_{1}*, *z_{2}
*=* *x*_{2}* + i*y_{2} so
that

as required. (we have employed, apart from Definition (1), the multiplication rule for complex numbers and known formulae of trigonometry.)

We note that the exponential function does not at
all vanish for any complex *z = x + iy. *In fact, |*e*^{z}|
= *e*^{x }> 0*.*

In particular, setting in (1) *x* = 0, *y*
= *j*, we obtain *Euler's
classical formula**

*L.Euler derived in his "Introduction to the analysis of the infinitely small"; formulae equivalent to (4) appeared in his work, starting in 1740.

With the aid of Euler's formula, any complex
number *z *with modulus *r* and argument *j *may be written in the exponential
form:

Side by side with the properties 1) - 4), which
are true in the real as well as the complex domain, the
exponential function of a complex variable has the specific
properties: It becomes *periodic *with purely imginary
basic period 2*p**i*. In
fact, for any integer *k*, we have

as follows from Euler's formula *e*^{2k}^{p }^{i}*
= *1.

On the other hand, if

then, by Definition (1),

whence *x*_{2} *= x*_{1}*, y*_{2}*
= y*_{1}* + *2*k**p**i*
or

where *k* is an integer.

Due to its periodic property, the study of the function *e*^{z}
throughout the entire plane reduces to its study in the strip .
That simple analysis shows that the mapping (1) has only one
sheet in the entire strip: Equality (7)yields and the strip contains
only one pair of points linked by these relations.

We now introduce in the *w*-plane polar co-ordinates by
setting *w* = *r**e*^{i}^{q}; then (1) yields the
two equations

Hence, (1) transforms the line *y* = *y*_{0}*
*into into the ray *q* = *y*_{0
}and the segment *x* = *x*_{0}*,*0 £ *y** <
*2*p**, *into teh circle the The
strip 0 < *y* < 2*p**
*thus becomes the *w*-plane with a cut along the
positive semi-axis and its half 0 < *y* < *p *into the upper half plane. In
general, the strip 0 < Im *z* < *h *of the
exponential function becomes the angle 0 < arg* w *<
*h* (Fig. 13).

*The **logarithmic
function** *is defined as the inverse of the
exponential function: The number *w* is the logarithm of
the number *z*, if *e*^{w} = *z*;
the notation is

This definition yields the basic property of the logarithm: If
*w*_{1} = ln *z*_{1}, *w*_{2}*
= *ln *z*_{2}, then ln *z*_{1 }+
ln *z*_{2 }is the logarithm of *z*_{1}
*z*_{2}:

In fact, we have consequently,

In particular, setting in (10) we find

The symbol arg *z* in (11) may denote any value of the
argument *z*, whence *every complex number **z *¹ 0 *has infinitely many logarithms. *In other
words, the logarithm is an infinitely-valued function: Its real
part is uniquely defined, its imaginary part exactly apart from a
term involving a multiple of 2*p*.* For the sake of clarity, we will denote this
multiple-valued function by the special symbol Ln *z*
so that

We will denote by the symbol ln *z* one of the values
of Ln *z*. If necessary, we will specially indicate which
particular value is involved, so that in all the receding
formulae, containing the symbol ln, we need not change the
notation.

* Euler arrived first at such an understanding of the logarithm; his idea was exposed in 1749 in "On the controversy between Bernoulli and Leibnitz regarding the logarithms of negative and imaginary numbers."

Consider at greater length the choice if the value Ln *z*.
Just as for the multi-valued functions considered above, the
value Ln *z* is determined by the argument, ascribed to
the point *z*. We assume that the point *z*
describes, starting from *z*_{0}* *¹ 0, some curve *C
*which does not pass through the origin of the co-ordinates.
Since, as above, we denote by arg *z *uniquely and
continuously along *C* the function Arg *z, *defined
by some fixed initial value arg *z*_{0}. We will
denote by ln *z* the value of Ln *z*, determined by
(11) for the chosen value of arg *z*; obviously, the
function ln *z* will be single-valued and continuous along
*C*.

Assume that the curve *C *is closed and does not
contain inside it the point *z* = 0. When *z*
describes *C*, the point *w *= ln *z*
describes some closed curve *G*;
other values of the logarithm, determined by other initial values
of arg *z*_{0}, describe curves *G*_{k}, differing
from *G* only by the
displacement on the vector 2*k**p**i,
k = ± *1*, ± *2, ··· (Fig. 14, solid lines). Now,
if is
a curve without points of self-intersection, containing *z = *0
inside it, then, for a complete circuit of its point *z*
in the positive direction, the point *w* = ln *z*
does not return to its initial position, but assumes the new
position *w*_{0}^{(1)}* *= *w*_{0}*
*+ 2*p**i* (Fig 14,
broken lines).

Hence, in any region *D*, which does not contain closed
curves, circumventing the point *z* = 0, one may select
infinitely many continuous and single-valued *branches *of
the multi-valued function *w* = Ln *z*, the values
of which at each fixed point differ from each other only by the
term 2*k**p**i*.
Every such branch ln *z* will generate a mutually
single-valued mapping of the region *D* and, consequently,
by the theorem regarding the derivative of the inverse function,
have the derivative

(Note that the derivative is one and the same for all
branches.) Thus, all such branches of Ln *z* will be
analytic functions.

However, if the region *D* contained one closed curve,
surrounding the point *z* = 0 (for example, if it were to
contain this point inside it), then in such a region the branches
of the function Ln *z* cannot be separated from each
other. The point *z* = 0, at which all branches of Ln *z
*would* *join, is called the *branch point *of
this function.

**9.
Trigonometric and hyperbolicic functions**** **in
the complex domain are expressed simply by exponential functions.
For the real variable *x*, Euler's formula (4) of **8.****
**yields

whence

Thus, by definition and for any complex *z,*

Thus, the defined functions 1) coincide for real *z*
= *x *with the ordinary sine and cosine; 2) they are
everywhere analytic; 3) they obey the ordinary differentiation
formulae

4) they are periodic with the real period 2*p*; sin* z* is an odd , cos z
is an even function: 6) they obey the usual trigonometric
relations

All these assertions follow from Definition (1); the reader should verify this by undertaking the corresponding operations.

Consider the mapping, realized by the first of these functions. Setting

we find

We see that our mapping may be interpreted as a
superposition of mappings, which we have already studied. First
of all, we find the condition of its single-sheetedness. Let the
region *D* become by Mapping (2) successively *D*_{1},
*D*_{2}, *D*_{3}. The first and
third of Mappings (2) are single-sheeted everywhere; for this
property for the second, it is necessary and sufficient that *D*
does not contain a pair of points *z*_{1}' and *z*_{2}"
for which

where *k *¹ 0 is an integer (cf. (7) of **8.**) It is necessary and sufficient for the
single-sheetedness of (3) that *D*_{3} does not
contain pairs of points *z*_{3}' and *z*_{3}"
for which

(cf. (2) of **7.**). Going with the aid of (2) to the *z* -plane,
we find that it is necessary and sufficient for
single-sheetedness of the mapping *w* = sin *z* in
the domain *D* that *D* does not contain a pair of
points *z*'* *and *z*" for which, on
the one hand.

and, on the other hand, *e*^{i(z'
+ z")} = -1, or

For example, these conditions are satisfied by
the strip *-**p** *<*
x *< *p*, *y** *> 0. The successive stages of
these mappings are shown in Fig. 15. The family of rays *x*
= *x*_{0} and segments *y*_{0}*,
*respectively, become the family of confocal hyperbolae and
ellipses; twice more the strip -*p *<
*x* <*p, **y >
0 *becomes the upper half-plane.

We see that sin z is *unbounded *in the
complex plane; for example, on the rays *x* = ±*p*/2, *y* > 0, it assumes
real values with modulus larger than unity and, generally
speaking, arbitrarily large.

We note yet that in the (closed) half-plane -*p *£ *x *£ *p*, *y* > 0, it assumes real values with modulus
larger than unity and, generally speaking, arbitrarily large.

Note yet that, in the (closed )
half-plane -*p *£ *x *£ *p*, *y** *³ 0, the function sin *z* assumes the value 0 only
at the points *z* = 0 and *z = *±*p*; taking into consideration the
unevenness and periodicity of this function, one may conclude
that it becomes 0 only on the real axis at the points

In order to complete this work, we go to Fig. 16 of the *surface
of the modulus *or the* pattern* of the function sin *z*,
i.e., the surface in the (*x*, *y*, *u*)
space with the equation *u* = |sin *z*|; this is
the ;periodic surface with the real period *p.**
*There are plotted on it two systems of lines - the level
lines of |sin *z*| and arg sin *z*. The section of
the vertical plane through the *x*-axis yields the graph
of |sin *x*|*. As we move away from this axis, the surface
rises, while ordinates of its points grow rapidly - the surface
approaches the cylinder *u* = ½ *e*^{|y|}*.*

*) The intersections of the surfaces *x = k**p *and *x*
= (2*k* + 1) (*k *= 0, ±1*, *±2, ··· )
yield the graphs of the hyperbolic functions |sinh *y*|
and |cosh *y*| with which we will become acquainted soon.
Fig. 16 shows parts of those intersections *x* = 0 and *x*
= 3*p*/2; however, we may assume that there are in this figure two
co-ordinate origins - *O *relates to sin and sinh, but *O*_{1}*
*to cos and cosh.

The reflection, yielding the function cos *z*
by the relation

differs from the considered one by shear.

The functions tan *z* and cotan *z *are defined
by

The function tan *z* is analytic
everywhere except at the points where cos *z *vanishes,
i.e., as follows from the preceding discussion, everywhere except
at the points *z*_{k} =* **p*/2 + *k*p*
*(*k *= 0, ±1, ±2, ···); on approaching these
points, tan *z* increases beyond all bounds. The same
applies to the function cotan *z* and the points *z*_{k}
= *k**p *(*k *=
0, ±1, ±2, ···).

It follows from (6) that these functions are periodic with
period *p*. In fact, for
example,

We will study the mapping by *w* = tan *z*
in **33.** We
present here only the pattern of tan *z*, i.e., the
surface *u* = |tan *z*| (Fig. 17); this surface is
periodic with real period *p*/2. I
has clearly expressed peaks at the points z =
*p*/2 + *k**p *(*k
*= 0, ±1, ±2, ···); its intersection with the vertical
plane, passing through the *x*-axis, yields the graph |tan
*x*|*. Further away from this axis, the surface becomes
more plane and approaches the plane *u* = 1. The level
lines of |tan *z*| and arg tan *z* are plotted on
this surface.

* The intersections of the
surfaces with the planes *x* = *k**p *and *x*
= (2*k* + 1)*p*/2 (*k* = 0, ±1, ±2, ···) yield the graphs
of the hyperbolic functions |tanh *y*| and |cotanh *y*|,
respectively.

*Hyperbolic functions *in the complex
domain are given by

They are expressed very simply in terms of the trigonometric functions

and therefore differ little from them. Figs. 16 and 17 above
show the intersections of the surfaces of the moduli for sin *z*
and tan *z*, yielding the graphs of the hyperbolic
functions.

As we have seen, the trigonometric and hyperbolic functions
are expressed in terms of the exponential function, whence then
inverse trigonometric and inverse hyperbolic functions may be
expressed in terms of logarithms. For example, we obtain such an
expression for *w* = arccos *z*. We have, by
definition,

whence *e*^{2iw}* - *2*ze*^{iw}*
+ *1* = *0; solving the quadratic equation for *e*^{iw},
we find and

( the ± sign in the formula for the solution of the quadratic equation may be omitted if one understands the root as a double-valued function). By the relation , the change in the sign of the roots reduces to a change in sign of the logarithm, whence the sign (-) in the last formula my be omitted:

( we have agreed above that the root has two signs).

Analogous formulae may also be found for the other functions:

All these function are multi-valued, because ln on the right hand sides of (10) and (11) may denote any value of the logarithm. Special separation of their single-valued branches is analogous to that given above; all such branches will be analytic functions.