A linear function of a complex variable z is a function of the form where a and 6 are given complex numbers, and a Ф 0. A linear function is defined for all values ​​of the independent variable z, it is single-valued and, since the inverse function is also single-valued, it is univalent in the entire plane z. A linear function is analytic in the entire complex plane, and its derivative is therefore conformal to the mapping carried out by it in the entire plane. A linear-fractional function is a function of the form - given complex numbers, and a linear-fractional function is defined for all values ​​of the independent variable zy except z = -|, it is single-valued and, since the inverse function Elementary functions of a complex variable Fractional-rational functions Power function Exponential function Logarithmic function The trigonometric and hyperbolic functions are single-valued, univalent in the entire complex plane, excluding the point z = - In this region, the function (3) is analytic and its derivative, therefore, the mapping carried out by it is conformal. Let us extend the function (3) at the point z = - \, setting t) = oo, and put the point at infinity w = oo in correspondence with the point z(oo) = Then the linear-fractional function will be univalent in the extended complex plane z. Example 1. Consider a linear-fractional function From the equality it follows that the moduli of complex numbers r and u are related by the relation and these numbers themselves are located on the rays emanating from the point O and symmetrical about the real axis. In particular, the points of the unit circle |z| = 1 go to the points of the unit circle N = 1. In this case, the conjugate number is assigned to the complex number (Fig. 11). Note also that the function r0 = -g maps the point at infinity r - oo to zero r0 - 0. 2.2. Power function A power function where n is a natural number, is analytic in the entire complex plane; its derivative = nzn~] for η > 1 is non-zero at all points except z = 0. Writing w and z in formula (4) in exponential form, we obtain that From formula (5) it can be seen that the complex numbers Z\ and z2 such that where k is an integer go to one point w. Hence, for n > 1 mapping (4) is not univalent on the z-plane. The simplest example of a domain in which the mapping ri = zn is univalent is the sector where a is any real number. In the region (7), mapping (4) is conformal. - is multi-valued, because for each complex number z = r1v Φ 0, one can indicate n different complex numbers, such that their nth power is equal to z: Note that a polynomial of degree n of a complex variable z is a function where are given complex numbers, and ao Ф 0. A polynomial of any degree is an analytic function on the entire complex plane. 2.3. Fractional-rational function A fractional-rational function is a function of the form where) are polynomials of a complex variable z. A fractional rational function is analytic in the entire plane, except for those points where the denominator Q(z) vanishes. Example 3. The Zhukovsky function is analytic in the entire plane z, excluding the point z = 0. Let us find out the conditions on the region of the complex plane under which the Zhukovsky function considered in this region will be univalent. M Let the points Z) and zj be transferred by function (8) to one point. Then, for , we obtain that Hence, for the univalence of the Zhukovsky function, it is necessary and sufficient that the condition be satisfied. An example of a domain that satisfies the univalence condition (9) is the exterior of the circle |z| > 1. Since the derivative of the Zhukovsky function Elementary functions of a complex variable Fractional-rational functions Power function Exponential function Logarithmic function Trigonometric and hyperbolic functions are nonzero everywhere except at points, then the mapping of the area carried out by this function will be conformal (Fig. 13). Note that the interior of the unit disk |I is also the domain of univalence of the Zhukovsky function. Rice. 13 2.4. The exponential function We define the exponential function ez for any complex number z = x + y by the following relation: For x = 0 we obtain the Euler formula: Let us describe the main properties of the exponential function: 1. For real z this definition matches the usual. This can be verified directly by setting y = 0 in formula (10). 2. The function ez is analytic on the entire complex plane, and the usual differentiation formula is preserved for it. 3. The addition theorem is preserved for the function ez. Let 4. The function ez is periodic with imaginary main period 2xi. Indeed, for any integer k On the other hand, if then from definition (10) it follows that Whence it follows that, or where n is an integer. The strip does not contain a single pair of points related by relation (12), so it follows from the study that the mapping w = e" is univalent in the strip (Fig. 14). Since it is a derivative, this mapping is conformal. Remark niv. The function rg is univalent in any strip 2.5 Logarithmic function From the equation where given, the unknown, we obtain Hence Thus, the function inverse of the function is defined for any and is represented by the formula where This multi-valued function is called logarithmic and is denoted as follows denoted by Then for Ln z we obtain the formula 2.6 Trigonometric and hyperbolic functions From Euler's formula (11) for real y we obtain Whence We define the trigonometric functions sin z and cos z for any complex number z using the following formulas: The sine and cosine of a complex argument have interesting properties We list the main ones: The functions sinz and cos z: 1) for real x z -x coincide with the usual sines and cosines; 2) are analytic on the entire complex plane; 3) obey the usual differentiation formulas: 4) are periodic with a period of 2n; 5) sin z - odd function, a cos z - even; 6) the usual trigonometric relations are preserved. All listed properties are easily obtained from formulas (15). The functions tgz and ctgz in the complex domain are defined by formulas, and hyperbolic functions are defined by the formulas "Hyperbolic functions are closely related to trigonometric functions. This relationship is expressed by the following equalities: The sine and cosine of a complex argument have another important property: on the complex plane | \ take arbitrarily large positive Using properties 6 and formulas (18), we obtain that Elementary functions of a complex variable Fractional-rational functions Power function Exponential function Logarithmic function Trigonometric and hyperbolic functions Whence Assuming we have Example 4. It is easy to check that -4 ,

11 Basic functions of a complex variable

Recall the definition of the complex exponent - . Then

Maclaurin series expansion. The radius of convergence of this series is +∞, which means that the complex exponent is analytic on the entire complex plane and

(exp z)"=exp z; exp 0=1. (2)

The first equality here follows, for example, from the theorem on term-by-term differentiation of a power series.

11.1 Trigonometric and hyperbolic functions

The sine of a complex variable called a function

Cosine of a complex variable there is a function

Hyperbolic sine of a complex variable is defined like this:

Hyperbolic cosine of a complex variable-- is a function

We note some properties of the newly introduced functions.

A. If x∈ ℝ , then cos x, sin x, ch x, sh x∈ ℝ .

B. There is the following connection between trigonometric and hyperbolic functions:

cos iz=ch z; sin iz=ish z, ch iz=cos z; shiz=isinz.

B. Basic trigonometric and hyperbolic identities:

cos 2 z+sin 2 z=1; ch 2 z-sh 2 z=1.

Proof of the basic hyperbolic identity.

The main trigonometric identity follows from the Ononian hyperbolic identity when the connection between trigonometric and hyperbolic functions is taken into account (see property B)

G Addition Formulas:

In particular,

D. To calculate the derivatives of trigonometric and hyperbolic functions, one should apply the theorem on term-by-term differentiation of a power series. We get:

(cos z)"=-sin z; (sin z)"=cos z; (ch z)"=sh z; (sh z)"=ch z.

E. The functions cos z, ch z are even, while the functions sin z, sh z are odd.

G. (Periodicity) The function e z is periodic with period 2π i. The functions cos z, sin z are periodic with a period of 2π, and the functions ch z, sh z are periodic with a period of 2πi. Furthermore,

Applying the sum formulas, we get

W. Decompositions into real and imaginary parts:

If a single-valued analytic function f(z) bijectively maps a domain D onto a domain G, then D is called a domain of univalence.

AND. Domain D k =( x+iy | 2π k≤ y<2π (k+1)} для любого целого k является областью однолистности функции e z , которая отображает ее на область ℂ* .

Proof. Relation (5) implies that the mapping exp:D k → ℂ is injective. Let w be any nonzero complex number. Then, solving the equations e x =|w| and e iy =w/|w| with real variables x and y (we choose y from the half-interval )