I. Basic Concepts
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 + i0 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 x1 + iy1 and x2 + iy2 are equal,
if and only if x1 = x2 and y1 = y2.
However, note that if x2 = x1, but y2 = -y1, the complex number x2+ iy2 is said to be conjugate to x1 + iy1 and denoted by Thus
We now continue with the operations on complex numbers.
1) Addition: Sum z1 + z2 of the complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 is called the complex number
The stated definition yields at once the laws of addition:
a) Commutative Law: z1
+ z2 = z2 + z1.
b) Associative Law: z1 + (z2 + z3) = (z1 + z2) + z3.
If z1 and z2 are real numbers( i.e., y1 = y2 = 0, Definition (4) coincides with the definition of addition of real numbers.
Addition admits the inverse operation: One may likewise find for any two complex numbers z1 = x1 + iy1 , z2 = x2 + iy2 a number z so that z2 + z = z1. This number is called the difference of the numbers z1 and z2 and denoted by z1 - z2. Obviously,
2) Multiplication: Product z1z2 of the complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 is called the complex number
This definition yields the laws of multiplication:
a) Commutative Law: z1z2
b) Associative Law: z1(z2 z3) = (z1z2) z3.
c) Distributive Law: (z1 + z2)z3 = z1z3 + z2z3.
If z1 and z2 are real numbers, Definition (6) coincides with the ordinary definition. For z1 = z2 = 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 z2 ¹ 0; then one may find the number z such that z2 z = z1; by (6), one must solve for this the system of equations
which for z2 ¹ 0 is always uniquely soluble, since its determinant x2² + y2² > 0. This number z is called the quotient of the two numbers z1, z2 and is denoted by z1/z2. Solving System (9), we find
It is readily seen that (10) may be obtained by multiplying the numerator and denominator of the fraction z1/z2 by
3) Integral Power The product of n equal numbers z is called the n-th power of the number z and denoted by zn:
The inverse operation - taking the root - is defined as follows: The number w is called the n-th root of z, if wn = 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 i² = -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 z1, z2, z3", 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 z1 and z2 are represented, respectively, by the vectors in the directions of the diagonal of the parallelogram, formed by the vectors z1 and z2.
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 2p :
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 z1 = Arg z2.
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 z1 by z2, the vector z1 is stretched |z2| times and, in addition, it is rotated counter-clockwise by the angle arg z2. 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 = z1z2; in order to obtain z, it is sufficient to construct with the segment Oz1 as base the triangle Oz1z, similar to the triangle O1z2.
If |z2| < 1, then z1shrinks effectively 1/|z2| times.
Moreover, division of a complex number z1 by z2 reduces to multiplication of z1by 1/z2, 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 Ozz and Ozw 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 2p/n, because one may add to the value of arg z a multiple of 2p, 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 G0, G1, G2, two cuts g1, g2 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; G0 and g1 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 + isin a) and represent Function (5) by the compound function, involving the functions
a) z1 = (cos a + i sin a)z,
b) z2 = kz1,
c) w = z2 + b.
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 z1-plane with the similarity coefficient k. The mapping c) is geometrically a displacement of the entire z2 -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 z0 = x0 + i y0 except, may be, at the point z0.
We will say that the exists the limit of the function f(z) as z ® z0 ( 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 z0 (except, may be, z0 itself) the corresponding point w0 lies in an e-neighbourhood of w0; in other words, if there follows from the inequality
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 z0. In other words, if the limit exists, then as z tends to z0 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 z0 (including the point z0 itself) if it is defined in some neighbourhood of z0 (in cluding the point z0 itself) and
Obviously, it is necessary and sufficient for the continuity of f(z) at the point z0 that the functions u(x, y) and v(x, y) are continuous at the point (x0, y0).
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 z1 and z2 of , satisfying the inequality |z1 - z2| < 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 + ib. 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 s0 and n0 (i.e., complex numbers with moduli 1) and such that rotation of s0 and n0 is performed by a right counter-clockwise angle (i.e., n0 = is0). Using the fact that the calculation of the derivative does not depend on the direction, we obtain, taking the derivative once in the direction s0 , another time in the direction n0):
( 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 n0 = is0 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 s0 = 1, n0 = i, we obtain (9).
We will still present the d'Alembert-Euler conditions in polar co-ordinates (r, j). Let s0 be the unit vector of the tangent to the circle |z| = r in the counter-clockwise direction and n0 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 ez 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 = zn and w = z1/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(cosj + isinj), w = r(cosq + i sinq), 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 = zn, 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 z1 and z2 with different moduli and arguments, differing by an integral multiple 2p/n, and only such points, become for Mapping (1) a single point. Consequently, it is necessary and sufficient for a single sheet mapping w = zn in some region D that D does not contain either of the two points z1 and z2, linked by
This condition is satisfied, for example, by the sectors
each of which is transformed by the mapping w = zn 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.
equivalent to (1) yields for the straight lines u = u0 and v = v0 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 = zn is analytic in the entire plane, because there exists for any z
the inverse of the function z = wn , 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 z0 one of the values of the argument at the point z ¹ 0 and assume that the point z describes, for example, with z0 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 z0*. 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 z0, 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 z0. Knowledge of the root, determined by by another choice of the initial value arg z0 (differing from the preceding one by an integral multiple of 2p) for a complete circuit C, obviously, also describes closed curves Gk, differing from G only by a rotation by an angle 2kp/n. k = 1,2, ··· , n-1 (solid curves in Fig. 10 below).
Let be a closed curve without intersections, containing z = 0 inside, and z0 some point on the curve . Then, for a complete circuit of , starting at z0 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 w01, where
w0(1) =(cos2p/n+isin 2p/n) w0 - the value of other than w0.
This is explained by the fact that arg z undergoes during a circuit of the increase 2p. 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 2kp/n + isin 2kp/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 = zn.
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 z1 and z2 become the single point w; we will then have z1 + 1/ z1= z2 + 1/ z2, whence
Thus, it is necessary and sufficient for single-sheetedness if the mapping (1) at any point that D does not contain any two points z1 and z2, linked by the relation z1z2 = 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 + isin j), w = u + iv and separate the real and imaginary parts. The mapping (1) then becomes
and we see that every circle |z| = r0 < 1 becomes the curve
i.e., an ellipse with the semi-axes a = ½(r0 + 1/r0),b = ½(1/r0 - r0), passed in a negative direction*. As r0 1, this ellipse contracts into the segment [-1, 1] of the u-axis, for r0 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 = j0 , 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| = r0 > 1 map into the ellipses with semi-axes a = ½(r0 + 1/r0),b = ½(1/r0 - r0). These ellipses coincide with those into which map the circles |z| = r0 < 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 z1 and z2 , linked by the condition z1 z2 = 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 z1z2 = 1.
Denote by r 1,q1 and r2, q2, 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 (q1 + q2)/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, q1 (or q2) changes by 2p, but q2 (or q1) 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 q1 and q2 change by 2p and, consequently, the argument of the root (q1 + q2)/2 likewise changes by 2p. 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 q1 nor q2 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 z1z2 = 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 ez 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 zn = (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 zn = n arctan (y/n)/(1 + x/n); converting in the first expression the small of higher order term (x² + y²)/n² 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 |ez| = ex and arg ez = 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)
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 ez in the direction of the x-axis:
which also yields (2).
Finally, we set for the proof of 4) z1 = x1 + iy1, z2 = x2 + iy2 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, |ez| = ex > 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 2pi. In fact, for any integer k, we have
as follows from Euler's formula e2kp i = 1.
On the other hand, if
then, by Definition (1),
whence x2 = x1, y2 = y1 + 2kpi or
where k is an integer.
Due to its periodic property, the study of the function ez 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 = reiq; then (1) yields the two equations
Hence, (1) transforms the line y = y0 into into the ray q = y0 and the segment x = x0,0 £ y < 2p, into teh circle the The strip 0 < y < 2p 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 ew = z; the notation is
This definition yields the basic property of the logarithm: If w1 = ln z1, w2 = ln z2, then ln z1 + ln z2 is the logarithm of z1 z2:
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 2p.* 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 z0 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 z0. 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 z0, describe curves Gk, differing from G only by the displacement on the vector 2kpi, 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 w0(1) = w0 + 2pi (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 2kpi. 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
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 2p; 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 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 D1, D2, D3. 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 z1' and z2" 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 D3 does not contain pairs of points z3' and z3" 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, ei(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 = x0 and segments y0, 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 = kp and x = (2k + 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 = 3p/2; however, we may assume that there are in this figure two co-ordinate origins - O relates to sin and sinh, but O1 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 zk = p/2 + kp (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 zk = kp (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 + kp (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 = kp and x = (2k + 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 e2iw - 2zeiw + 1 = 0; solving the quadratic equation for eiw, 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.