Elementary fkp. Functions of a complex variable. Differentiation of functions of a complex variable. Cauchy-Riemann conditions. Basic functions of a complex variable

11 Basic functions of a complex variable

Let us recall the definition of a complex exponent – ​​. Then

Maclaurin series expansion. The radius of convergence of this series is +∞, which means that the complex exponential 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

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-- this is a function

Let us note some properties of the newly introduced functions.

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

B. The following connection exists between trigonometric and hyperbolic functions:

cos iz=ch z; sin iz=ish z, ch iz=cos z; sh iz=isin z.

B. Basic trigonometric and hyperbolic identities:

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

Proof of the main hyperbolic identity.

The main trigonometric identity follows from the main hyperbolic identity when taking into account the connection between trigonometric and hyperbolic functions (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, and the functions sin z, sin z are odd.

J. (Frequency) 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, sin z are periodic with a period of 2πi. Moreover,

Applying the sum formulas, we get

Z. Expansion 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 univalent domain.

AND. Region 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. From relation (5) it follows that the mapping exp:D k → ℂ is injective. Let w be any non-zero complex number. Then, solving the equations e x =|w| and e iy =w/|w| with real variables x and y (y is chosen from a half-interval for n > 1 is different from zero at all points except z = 0. Writing w and z in exponential form in formula (4), we obtain that From formula (5) it is clear that complex numbers Z\ and z2 such that where k is an integer, go to one point w. This means that for n > 1 the mapping (4) is not univalent on the plane z. The simplest example of a domain in which the mapping φ = zn is univalent is. sector where a is any real number. In domain (7), mapping (4) is conformally multivalued, since for each complex number z = ε1в Ф 0 one can specify 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 given complex numbers, and ao Φ 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) - polynomials of the complex variable z. The fractional rational function is analytic throughout the plane, except for those points at which 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 for 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 at we get that So, for the Zhukovsky function to be univalent, it is necessary and sufficient to satisfy the condition An example of a region 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 is nonzero everywhere except at points, the mapping of the domain 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. Exponential function We define the exponential function ez for any complex number z = x + y by the following relation: For x = 0 we obtain Euler's formula: Let us describe the main properties of the exponential function: 1. For real z, this definition coincides with the usual one. This can be verified directly by setting y = 0 in formula (10). 2. The function ez is analytic on the entire complex plane, and for it the usual differentiation formula is preserved. 3. For the function ez the addition theorem is preserved.