21.2 Number series (NR):
Let z 1 , z 2 ,…, z n be a sequence of complex numbers, where
ODA 1. An expression of the form z 1 +z 2 +…+z n +…=(1) is called a PD in the complex domain, and z 1 , z 2 ,…, z n are members of the number series, z n is the common member of the series.
ODA 2. The sum of the first n terms of the complex PD:
S n \u003d z 1 + z 2 + ... + z n is called nth partial sum this row.
ODA 3. If there is a finite limit for n sequences of partial sums S n of a numerical series, then the series is called converging, while the number S itself is called the sum of the PD. Otherwise, the CR is called divergent.
The study of the convergence of PRs with complex terms is reduced to the study of series with real members.
Required sign of convergence:
converges
Def4. CR is called absolutely convergent, if a series of moduli of terms of the original PD converges: |z 1 |+|z 2 |+…+| z n |+…=
This series is called modular, where |z n |=
Theorem(on the absolute convergence of the CR): if the modular series , then the series also converges.
In the study of the convergence of series with complex members, all known sufficient criteria for the convergence of sign-positive series with real members are used, namely, comparison criteria, d'Alembert, radical and integral Cauchy criteria.
21.2 Power series (SR):
Def5. SR in the complex plane is called an expression of the form:
c 0 +c 1 z+c 2 z 2 +…+c n z n =, (4) where
c n - coefficients SR (complex or real numbers)
z=x+iy – complex variable
x, y are real variables
Also consider SR of the form:
c 0 +c 1 (z-z 0)+c 2 (z-z 0) 2 +…+c n (z-z 0) n +…=,
Which is called SR in powers of difference z-z 0 , where z 0 is a fixed complex number.
def 6. The set of z values for which CP converges is called convergence region SR.
OP 7. A CP converging in some region is called absolutely (conditionally) convergent if the corresponding modular series converges (diverges).
Theorem(Abel): If the SR converges for z=z 0 ¹0 (at the point z 0), then it converges, and, moreover, absolutely for all z satisfying the condition: |z|<|z 0 | . Если же СР расходится при z=z 0 ,то он расходится при всех z, удовлетворяющих условию |z|>|z 0 |.
It follows from the theorem that there exists a number R called convergence radius SR, such that for all z for which |z|
The region of convergence of SR is the interior of the circle |z| If R=0, then СР converges only at the point z=0. If R=¥, then the region of convergence of SR is the entire complex plane. The area of convergence of the CP is the interior of the circle |z-z 0 | The convergence radius of the SR is determined by the formulas: 21.3 Taylor series: Let the function w=f(z) be analytic in the circle z-z 0 f(z)= =C 0 +c 1 (z-z 0)+c 2 (z-z 0) 2 +…+c n (z-z 0) n +…(*) whose coefficients are calculated by the formula: cn=, n=0,1,2,… Such a SR (*) is called a Taylor series for the function w=f(z) in powers of z-z 0 or in the vicinity of the point z 0 . Taking into account the generalized Cauchy integral formula, the coefficients of the Taylor series (*) can be written as: C is a circle centered at the point z 0 , completely lying inside the circle |z-z 0 | For z 0 =0, the series (*) is called near Maclaurin. By analogy with expansions in the Maclaurin series of the main elementary functions of a real variable, one can obtain expansions of some elementary FKPs: Expansions 1-3 are valid on the entire complex plane. 4). (1+z) a = 1+ 5). log(1+z) = z- Expansions 4-5 are valid in the region |z|<1. Let us substitute the expression iz in the expansion for e z instead of z: (Euler formula) 21.4 Laurent series: Series with negative degrees of difference z-z 0: c -1 (z-z 0) -1 +c -2 (z-z 0) -2 +…+c -n (z-z 0) -n +…=(**) By substitution, the series (**) turns into a series in powers of the variable t: c -1 t+c -2 t 2 +…+c - n t n +… (***) If the series (***) converges in the circle |t| We form a new series as the sum of the series (*) and (**) by changing n from -¥ to +¥. …+c - n (zz 0) - n +c -(n -1) (zz 0) -(n -1) +…+c -2 (zz 0) -2 +c -1 (zz 0) - 1 +c 0 +c 1 (zz 0) 1 +c 2 (zz 0) 2 +… …+c n (z-z 0) n = (!) If the series (*) converges in the region |z-z 0 | Let the function w=f(z) be analytic and single-valued in the annulus (r<|z-z 0 | whose coefficients are determined by the formula: C n = (#), where C is a circle centered at the point z 0 , which lies completely inside the ring of convergence. The row (!) is called near Laurent for the function w=f(z). The Laurent series for the function w=f(z) consists of 2 parts: The first part f 1 (z)= (!!) is called right part Laurent row. The series (!!) converges to the function f 1 (z) inside the circle |z-z 0 | The second part of the Laurent series f 2 (z)= (!!!) - main part Laurent row. The series (!!!) converges to the function f 2 (z) outside the circle |z-z 0 |>r. Inside the ring, the Laurent series converges to the function f(z)=f 1 (z)+f 2 (z). In some cases, either the main or regular part of the Laurent series may either be absent or contain a finite number of terms. In practice, to expand a function in a Laurent series, the coefficients C n (#) are usually not calculated, because it leads to cumbersome calculations. In practice, proceed as follows: one). If f(z) is a fractional-rational function, then it is represented as a sum of simple fractions, while a fraction of the form , where a-const is expanded into a series of geometric progression using the formula: 1+q+q 2 +q 3 +…+=, |q|<1 The fraction of the species is expanded into a series, which is obtained by differentiating the series of a geometric progression (n-1) times. 2). If f(z) is irrational or transcendental, then the well-known Maclaurin expansions of the basic elementary FCFs are used: e z , sinz, cosz, ln(1+z), (1+z) a . 3). If f(z) is analytical at the point at infinity z=¥, then by substituting z=1/t the problem is reduced to the expansion of the function f(1/t) in a Taylor series in the neighborhood of the point 0, while the z-neighbourhood of the point z=¥ the exterior of a circle is considered with the center at the point z=0 and the radius equal to r (possibly r=0). L.1 DOUBLE INTEGRAL IN DECATIC COORD. 1.1 Basic concepts and definitions 1.2 Geometrical and physical meaning of DWI. 1.3 main properties of DWI 1.4 Calculation of DWI in Cartesian coordinates L.2 DWI in POLAR COORDINATES. CHANGING VARIABLES in DWI. 2.1 Change of variables in DWI. 2.2 DWI in polar coordinates. L.3Geometric and physical applications of DWI. 3.1 Geometric applications of DWI. 3.2 Physical applications of double integrals. 1. Mass. Calculation of the mass of a flat figure. 2. Calculation of static moments and coordinates of the center of gravity (center of mass) of the plate. 3. Calculation of the moments of inertia of the plate. L.4THE TRIPLE INTEGRAL 4.1 THREE: basic concepts. The existence theorem. 4.2 Basic SUT properties 4.3 SUT calculation in Cartesian coordinates L.5 CURVILINEAR INTEGRALS OVER COORDINATES OF THE II TYPE - KRI-II 5.1 Basic concepts and definitions of CWI-II, existence theorem 5.2 Key features of CWI-II 5.3 Calculation of KRI - II for various forms of setting the arc AB. 5.3.1 Parametric specification of the integration path 5.3.2. Explicit specification of the integration curve L. 6. CONNECTION BETWEEN DWI and KRI. SV-VA KRI II-th KIND ASSOCIATED WITH THE FORM OF THE WAY INTEGR. 6.2. Green's formula. 6.2. Conditions (criteria) for the contour integral to be equal to zero. 6.3. Conditions for the independence of the KRI from the form of the integration path. L. 7Conditions for the independence of the KRI of the 2nd kind from the form of the integration path (continued) K.8 Geometrical and physical applications of CWI of the 2nd kind 8.1 Calculation S of a flat figure 8.2 Calculation of work by change of force L.9 Surface integrals over surface area (SVI-1) 9.1. Basic concepts, existence theorem. 9.2. Main properties of PVI-1 9.3 Smooth surfaces 9.4. Calculation of PVI-1 by seeing to DVI. L.10. SURFACE INTEGRALS over COORD. (PVI2) 10.1. Classification of smooth surfaces. 10.2. PVI-2: definition, existence theorem. 10.3. Main properties of PVI-2. 10.4. PVI-2 calculation Lecture 11 11.1 Ostrogradsky-Gauss formula. 11.2 Stokes formula. 11.3. Application of PVI to the calculation of volumes of bodies. LK.12 ELEMENTS OF FIELD THEORY 12.1 Theor. Fields, osn. Concepts and definitions. 12.2 Scalar field. L. 13 VECTOR FIELD (VP) AND ITS CHARACTERISTICS.
13.1 Vector lines and vector surfaces. 13.2 Vector flow 13.3 Divergence of the field. Ostr.-Gauss formula. 13.4 Field circulation 13.5 Rotor (vortex) of the field. L.14 SPEC. VECTOR FIELDS AND THEIR CHARACTERISTICS 14.1 Vector 1st order differential operations 14.2 Vector differential operations II - order 14.3 Solenoid vector field and its properties 14.4 Potential (irrotational) IP and its properties 14.5 Harmonic field L.15 ELEMENTS OF THE FUNCTION OF A COMPLEX VARIABLE. COMPLEX NUMBERS (C/H). 15.1. K / h definition, geometric image. 15.2 Geometric representation of c/h. 15.3 Operation on c/h. 15.4 The concept of an extended complex z-pl. L.16 LIMIT OF A SEQUENCE OF COMPLEX NUMBERS. The function of a complex variable (FCF) and its limits. 16.1.
A sequence of complex numbers is a definition, a criterion for existence. 16.2
Arithmetic properties of aisles of complex numbers. 16.3
Function of a complex variable: definition, continuity. L.17 Basic elementary functions of a complex variable (FCV) 17.1.
Single-valued elementary FKPs. 17.1.1. Power function: ω=Z n . 17.1.2. Exponential function: ω=e z 17.1.3. Trigonometric functions. 17.1.4. Hyperbolic functions (shZ, chZ, thZ, cthZ) 17.2.
Multi-valued FKP. 17.2.1. Logarithmic function 17.2.2. arcsin number Z rev. number ω, 17.2.3.Generalized exponential function L.18FKP differentiation. Analytical function 18.1. Derivative and differential of FKP: basic concepts. 18.2. FKP differentiability criterion. 18.3. Analytic function L. 19 FKP INTEGRAL CALCULATION. 19.1 Integral from FKP(IFKP): def., reduction of KRI, teor. creatures. 19.2 About beings. IFKP 19.3 Theor. Cauchy L.20. The geometric meaning of the modulus and argument of the derivative. The concept of conformal mapping. 20.1 Geometric meaning of the modulus of the derivative 20.2 Geometric meaning of the derivative argument L.21. Series in the complex domain. 21.2 Number series (NR) 21.2 Power series (SR): 21.3 Taylor series Let a sequence of complex numbers be given z n = x n+ + it/ n , n= 1,2,... Numerical series
is called an expression of the form Numbers 21,2-2,... are called members of the series. We note that expression (19.1), generally speaking, cannot be considered as a sum, since it is impossible to perform the addition of an infinite number of terms. But if we confine ourselves to a finite number of terms in the series (for example, take the first P terms), then you get the usual amount that can actually be calculated (whatever the P). Sum of 5 first and members of the series is called n-th partial (private) sum of the series: Series (19.1) is called converging, if there is a finite limit n-x partial sums at P-? oo, i.e. exists The number 5 is called the sum of the series. If lirn S n does not exist or is equal to oc, then the series (19.1) is called divergent. The fact that the series (19.1) converges and its sum is equal to 5 can be written as This entry does not mean that all members of the series were added (it is impossible to do this). At the same time, by adding a sufficiently large number of terms of the series, one can obtain partial sums that deviate arbitrarily little from S. The following theorem establishes a connection between the convergence of a series with complex terms z n = x n + iy n and series with real members x n and at i. Theorem 19.1. For the convergence of the series (19.1) necessary and to enough, to meet two rows ? x n and? With valid P=1 them in yen. However, for equality ? z n = (T + ir and enough to ? x n = Proof. Let us introduce the notation for partial sums of series: Then S n = o p + ir n. Let us now use Theorem 4.1 from Section 4: so that the sequence S n = + ir n had a limit S == sg + ir, it is necessary and sufficient that the sequence(and(t p ) had a limit, and liiri = oh, lim t p = t. From here and p-yus l->oo the necessary statement blows, since the existence of limits of sequences (Sn), {(7
n ) and (t n ) is equivalent to the convergence of the series OS"OS"OS" ? Z n , ? X p and? y n respectively. L \u003d 1 L \u003d 1 P \u003d 1 With the help of Theorem 19.1, many important properties and statements that are valid for series with real terms can be immediately transferred to series with complex terms. Let's list some of these properties. 1°. Necessary sign of convergence. If a row? z n converges, then lim z n= 0. (The converse is not true: since lim z n = l-yuo i->oo 0 doesn't follow that row? z n converges.) 2°. Let the ranks? z n and? w n with complex terms converge and their sums are equal S and O respectively. Then a row? (z n+ w n) too converges and its sum is S + O. 3°. Let the row ]? z n converges and its sum is S. Then for any complex number L series? (A zn) also converges and its sum 4°. If we discard or add a finite number of terms to a convergent series, then we also get a convergent series. 5°. Cauchy convergence criterion. For the convergence of the series? z n necessary and sufficient that for any number e > 0 there was such a number N(depending on e) that for all n > N and for all R^ 0 ^2
zk Just as for series with real members, the concept of absolute convergence is introduced. Row z n called absolutely convergent, if the series converges 71
-
1
composed of modules of members of a given series %2
z n Theorem 19.2. If the series ^2 converges|*p|" then the series ^2z nalso converges. (In other words, if a series converges absolutely, then it converges.) Proof. Since the Cauchy convergence criterion is applicable to series with arbitrary complex terms, it applies, in particular, to series with real members. Take- meme arbitrary e> 0. Since the series JZ I z„| converges, then due to the tolerating Cauchy applied to this series, there is a number N, that for all P > N and for all R ^ 0 In § 1 it was shown that z+w^ |h| + |w| for any complex numbers z and w; this inequality easily extends to any finite number of terms. So So for any e> 0 there is a number N, such that for all P > So for any e> 0 there is a number N, such that for all P > > N and for all R^ 0 J2 z k but according to the Cauchy criterion, the series Y2 z n converges, which was to be proved. It is known from the course of mathematical analysis (see, for example, or )) that the statement converse to Theorem 19.2 is false even for series with real members. Namely, the convergence of a series does not imply its absolute convergence. Row J2 r p called conditionally convergent, if this series converges - Xia, but a row ^2 z n i composed of modules of its members diverges. Row z n is next to the real non-negative mi members. Therefore, the convergence criteria known from the course of mathematical analysis are applicable to this series. Let us recall some of them without proof. Signs of comparison. Let the numbers z u and w n, starting from some number N, satisfy the inequalities z n^ |w n |, n = = N, N+ 1,... Then: 1) if row ^2|w n | converges, then the series z n converges: 2) if the series ^2 S diverges, then the series ^2 1 w "1 diverges. Sign of d'Alembert. Let there be a limit Then: if I 1, then the series Y2 z n converges absolutely:
if I > 1, then the series ^2 z n diverges. At / = 1 “Radical” sign of Cauchy. Let it exist limit lim /zn = /. Then: if I 1, then the series z n converges absolutely;
if I > 1, then the row 5Z z n diverges. At I = 1 the sign does not answer the question about the convergence of the series. Example 19.3. Investigate convergence of series Solved and e. a) By definition of cosine (see (12.2)) So 00 1 (e p Let's apply the d'Alembert test to the series Y1 about(O) : Hence, the series ^ - (-) diverges. (The divergence of this series follows n= 1 2 " 2 " also from the fact that its terms do not tend to zero and, consequently, the necessary condition for convergence is not satisfied. You can also use the fact that the terms of the series form a geometric progression with denominator q\u003d e / 2\u003e 1.) On the basis of comparison, the series 51 0p so is the expense. b) Let us show that the quantities cos(? -f P) limited to the same number. Really, | cos (g 4- P)= | cos i cos n-sin i sin7i| ^ ^ | cos i|| cos 7?| 4-1 sing|| sin7?.| ^ | cosy| 4-1 sini| = A/, where M is a positive constant. From here Series 5Z converges. So, by comparison, the series cos (i 4" ii) also converges. Therefore, the original row 51 - ~^t 1 -~ converges ft-1 2 ” absolutely. Row 5Z z ki derived from row 51 zk discarding the first P k \u003d n + 1 k=1 members is called remainder (n-th remainder) row 51 zk- When convergence is also called the sum It is easy to see that 5 =
5 „ + g „, where 5 is the sum, a S n - partial sum row ^ Zf(- It immediately follows from this that if the series converges, then his n-th residue tends to the bullet at n-> oo. Indeed, let row Y2 z k converges, i.e. lirn 5n = 5. Then lim r n = lim (5 - 5n) =
ft-I
P->00 P->00 "->00 The existence of the concept of the limit of a sequence (1.5) allows us to consider series in the complex domain (both numerical and functional). Partial sums, absolute and conditional convergence of numerical series are standardly defined. Wherein the convergence of a series implies the convergence of two series, one of which consists of the real and the other of the imaginary parts of the terms of the series: For example, the series converges absolutely, and the series If the real and imaginary parts of a series converge absolutely, then the row, because the absolute convergence of the real and imaginary parts follows: Similarly to functional series in the real domain, complex functional series, the area of their pointwise and uniform convergence. Without change formulated and proven Weierstrass sign uniform convergence. are saved all properties of uniformly convergent series. In the study of functional series, of particular interest are power
ranks: , or after replacing : . As in the case of real variable, true abel theorem
: if the (last) power series converges at point ζ 0 ≠ 0, then it converges, and absolutely, for any ζ satisfying the inequality In this way, convergence region D this power series is a circle of radius R centered at the origin, where R − radius of convergence
− exact upper bound of values (Where did this term come from). The original power series will, in turn, converge in a circle of radius R with the center at z 0 . At the same time, in any vicious circle the power series converges absolutely and uniformly (the last statement immediately follows from the Weierstrass test (see the course “Series”)). Example .
Find the circle of convergence and examine for convergence in tt. z 1 and z 2 power series If at all boundary points the series converges absolutely or diverges according to the necessary criterion, then this can be established immediately for the entire boundary. To do this, substitute in a row from modules of terms value R instead of an expression and examine the resulting series. Example. Consider the series from the last example, changing one factor: The region of convergence of the series remains the same: resulting radius of convergence: If we denote the sum of the series by f(z), i.e. f(z) = (naturally, in region of convergence), then this series is called near taylor
functions f(z) or expansion of the function f(z) in a Taylor series. In a particular case, for z 0 = 0, the series is called near Maclaurin
functions f(z) . 1.7 Definition of basic elementary functions. Euler formula. Consider a power series If z is a real variable, then it represents is the Maclaurin series expansion of the function and, therefore, satisfies characteristic property of the exponential function: , i.e. Definition 1. Functions are defined similarly Definition 2. All three series converge absolutely and uniformly in any bounded closed region of the complex plane. From the three formulas obtained, a simple substitution deduces Euler formula:
From here it immediately follows demonstration
notation of complex numbers: Euler's formula establishes a connection between ordinary and hyperbolic trigonometry. Consider, for example, the function: Examples. Represent these expressions in the form 2. 4. Find linearly independent solutions of a linear DE of the 2nd order: The roots of the characteristic equation are: Since we are looking for real solutions to the equation, we can take the functions Let us define, in conclusion, the logarithmic function of a complex variable. As in the real domain, we will consider it inverse to the exponential one. For simplicity, we consider only the exponential function, i.e. solve the equation for w, which we call the logarithmic function. To do this, we take the logarithm of the equation, presenting z in exponential form: If instead of arg z write Arg z(1.2), then we obtain an infinite-valued function 1.8 Derivative of FKP. Analytic functions. Cauchy–Riemann conditions. Let w = f(z) is a single-valued function defined in the domain . Definition 1. derivative
from function f (z) at the point is called the limit of the ratio of the increment of the function to the increment of the argument, when the latter tends to zero: A function that has a derivative at a point z, is called differentiable
at this point. Obviously, all arithmetic properties of derivatives are satisfied. Example .
Using the Newton binomial formula, it is similarly deduced that The series for the exponent, sine, and cosine satisfy all the conditions for term-by-term differentiation. By direct verification it is easy to obtain that: Comment. Although the definition of the FKP derivative formally completely coincides with the definition for the FDP, it is, in essence, more complicated (see the remark in Section 1.5). Definition 2. Function f(z) , continuously differentiable at all points of the domain G, is called analytical
or regular
in this area. Theorem 1 .
If the function f (z) differentiable at all points of the domain G, then it is analytic in this area. (b/d) Comment. In fact, this theorem establishes the equivalence of regularity and differentiability of FKP on domains. Theorem 2. A function that is differentiable in some domain has infinitely many derivatives in that domain. (b/d. Below (in Section 2.4) this assertion will be proved under certain additional assumptions) We represent the function as the sum of the real and imaginary parts: Theorem 3. ( Cauchy − Riemann conditions).
Let the function f (z) is differentiable at some point . Then the functions u(x,y) and v(x,y) have partial derivatives at this point, and And called Cauchy–Riemann conditions
. Proof .
Since the value of the derivative does not depend on the way the quantity tends To zero, we choose the following path: We get: Similarly, when The converse is also true: Theorem 4. If functions u (x,y) and v(x,y) have continuous partial derivatives at some point that satisfy the Cauchy–Riemann conditions, then the function itself f(z) is differentiable at this point. (b/d) Theorems 1 – 4 show the fundamental difference between the FKP and the FDP. Theorem 3 allows you to calculate the derivative of a function using any of the following formulas: At the same time, one can consider X and at arbitrary complex numbers and calculate the derivative using the formulas: Examples. Check the function for regularity. If the function is regular, calculate its derivative. 19.4.1. Numerical series with complex terms. All basic definitions of convergence, properties of convergent series, convergence criteria for complex series do not differ in any way from the real case. 19.4.1.1. Basic definitions. Let an infinite sequence of complex numbers be given z
1 ,
z
2 ,
z
3 ,
…,
z
n
, … .The real part of the number z
n
we will denote a
n
, imaginary - b
n
(those. z
n
= a
n
+ i
b
n
,
n
= 1, 2, 3, …). Number series- type record. Partialamountsrow:
S
1
= z
1 ,
S
2
= z
1
+ z
2 ,
S
3
= z
1
+ z
2
+ z
3 ,
S
4
= z
1
+ z
2
+ z
3
+ z
4 ,
…, S
n
= z
1
+ z
2
+ z
3
+ … + z
n
,
… Definition. If there is a limit S
sequences of partial sums of the series for Find the real and imaginary parts of the partial sums: S
n
= z
1
+ z
2
+ z
3
+ … + z
n
= (a
1
+
i
b
1)
+ (a
2
+ i
b
2)
+ (a
3
+
i
b
3)
+ … + (a
n
+ i
b
n
)
= (a
1
+
a
2
+
a
3
+…+
a
n
)
+ Where symbols Example. Investigate for convergence series Let's write out several values of the expression Definition. Row Just as for numerical real series with arbitrary terms, it is easy to prove that if the series converges Row Example. Investigate for convergence series Let's make a series of modules (): 19.4.
1
.
3
. Properties of convergent series. For convergent series with complex terms, all properties of series with real terms are true: A necessary criterion for the convergence of a series.
The common term of the convergent series tends to zero as
If the series converges If the series converges, then the sum of its remainder aftern
-th term tends to zero at
If all terms of a convergent series are multiplied by the same numberWith
, then the convergence of the series is preserved, and the sum is multiplied byWith
.
Convergent rows (A
) and (V
) can be added and subtracted term by term; the resulting series will also converge, and its sum is equal to
If the terms of the convergent series are grouped arbitrarily and a new series is made up of the sums of the terms in each pair of parentheses, then this new series will also converge, and its sum will be equal to the sum of the original series. If a series converges absolutely, then for any permutation of its terms, the convergence is preserved and the sum does not change. If the rows (A
) and (V
) converge absolutely to their sums
Definition: Number series of complex numbers z 1, z 2, …, z n , … is called an expression of the form z 1 + z 2 + …, z n + … = ,(3.1) where z n is called the common term of the series. Definition: Number S n \u003d z 1 + z 2 + ..., z n is called the partial sum of the series. Definition: Series (1) is called convergent if the sequence (S n ) of its partial sums converges. If the sequence of partial sums diverges, then the series is called divergent. If the series converges, then the number S = is called the sum of the series (3.1). z n = x n + iy n, then series (1) is written as = + .
Theorem: Series (1) converges if and only if the series and , composed of the real and imaginary parts of the terms of series (3.1), converge. This theorem allows us to transfer the convergence criteria next to real terms to series with complex terms (necessary criterion, comparison criterion, d'Alembert, Cauchy criterion, etc.). Definition. The series (1) is called absolutely convergent if the series consisting of the modules of its members converges. Theorem. For the absolute convergence of the series (3.1), it is necessary and sufficient that the series and converge absolutely. Example 3.1. Find out the nature of the convergence of the series Solution. Consider the series Let us show that these series converge absolutely. To do this, we prove that the series Converge. Since , instead of a row, we take a row. If the last series converges, then the series also converges by comparison. The convergence of the series and is proved with the help of an integral criterion. This means that the series and converge absolutely and, according to the last theorem, the original series converges absolutely. 4. Power series with complex terms. Abel's power series theorem. Circle and radius of convergence. Definition. A power series is a series of the form where …, are complex numbers, called coefficients of the series. The region of convergence of the series (4.I) is the circle . To find the convergence radius R of a given series containing all powers, one of the formulas is used: If the series (4.1) does not contain all the powers of , then to find it, one must directly use the d'Alembert or Cauchy test. Example 4.1. Find the circle of convergence of the series: Solution: a) To find the radius of convergence of this series, we use the formula In our case Hence, the circle of convergence of the series is given by the inequality b) To find the radius of convergence of the series, we use the d'Alembert test. To calculate the limit, the L'Hopital rule was used twice. According to the d'Alembert test, the series will converge if . Hence we have the circle of convergence of the series . 5. Demonstrative and trigonometric functions complex variable. 6. Euler's theorem. Euler formulas. The exponential form of a complex number. 7. Addition theorem. Periodicity of the exponential function. The exponential function and trigonometric functions and are defined as the sums of the corresponding power series, namely: These functions are related by the Euler formulas: called, respectively, the hyperbolic cosine and sine, are related to the trigonometric cosine and sine by the formulas The functions , , , are defined as in the real analysis. For any complex numbers and the addition theorem holds: Any complex number can be written in exponential form: is his argument. Example 5.1. Find Solution. Example 5.2. Express the number in exponential form. Solution. Find the modulus and argument of this number: Then we get 8. Limit, continuity and uniform continuity of functions of a complex variable. Let E is some set of points in the complex plane. Definition. They say that on the set E function is given f complex variable z, if every point z E by rule f one or more complex numbers are assigned w(in the first case, the function is called single-valued, in the second - multi-valued). Denote w = f(z). E is the domain of the function definition. any function w = f(z) (z = x + iy) can be written in the form f(z) = f(x + iy) = U(x, y) + iV(x, y). U(x, y) = R f(z) is called the real part of the function, and V(x, y) = Imf(z) is the imaginary part of the function f(z). Definition. Let the function w = f(z) is defined and unique in some neighborhood of the point z 0 , excluding, perhaps, the very point z0. The number A is called the limit of the function f(z) at the point z0, if for any ε
> 0, one can specify a number δ > 0 such that for all z = z0 and satisfying the inequality |z – z 0 |< δ
, the inequality | f(z) – A|< ε.
write down It follows from the definition that z→z0 arbitrarily. Theorem. For the existence of the limit of the function w = f(z) at the point z 0 = x 0 + iy 0 it is necessary and sufficient that the limits of the function U(x, y) and V(x, y) at the point (x0, y0). Definition. Let the function w = f(z) is defined and unique in some neighborhood of the point z 0 , including this point itself. Function f(z) is called continuous at the point z 0 if Theorem. For continuity of a function at a point z 0 = x 0 + iy 0 it is necessary and sufficient that the functions U(x, y) and V(x, y) at the point (x0, y0). It follows from the theorems that the simplest properties related to the limit and continuity of functions of real variables carry over to functions of a complex variable. Example 7.1. Separate the real and imaginary parts of the function. Solution. In the formula that defines the function, we substitute To zero in two different directions, the function U(x, y) has different limits. This means that at the point z = 0 function f(z) has no limit. Next, the function f(z) defined at the points where . Let z 0 = x 0 + iy 0, one of these points. This means that at the points z = x + iy at
y 0 the function is continuous. 9. Sequences and series of functions of a complex variable. Uniform convergence. Power series continuity. The definition of a convergent sequence and a convergent series of functions of a complex variable of uniform convergence, corresponding to the theory of equal convergence, continuity of the limit of the sequence, the sum of the series are formed and proved in the same way as for sequences and series of functions of a real variable. Let us present the facts necessary for what follows concerning functional series. Let in the area D a sequence of single-valued functions of the complex variable (fn (z)) is defined. Then the symbol: called functional range. If z0 belongs D fixed, then the series (1)
will be numeric. Definition. Functional range (1)
is called convergent in the region D, if for any z owned D, the number series corresponding to it converges. If the row (1)
converges in the region D, then in this region one can define a single-valued function f(z), whose value at each point z owned D is equal to the sum of the corresponding number series. This function is called the sum of the series (1)
in the area of D . Definition. If for anyone z owned D, the following inequality holds: then the row (1)
is called uniformly convergent in the region D.ROWS
Number series
− diverges (due to the imaginary part).
. The converse is also true: from the absolute convergence of the complex series
Solution. 
region of convergence − circle of radius R= 2 with center in t. z 0 = 1 − 2i
. z 1 lies outside the circle of convergence and the series diverges. At , i.e. the point lies on the boundary of the circle of convergence. Substituting it into the original series, we conclude:
− the series converges conditionally according to the Leibniz criterion.
Substitute in a series of modules
. This is the basis for determining exponential function in the complex area:
.
The rest of the relations are obtained similarly. So:
(the expression in brackets is a number i
, written in exponential form)
we have: 
, which proves the theorem.
, which is a proper complex number, then the series is said to converge; number S
called the sum of the series and write S
= z
1
+ z
2
+ z
3
+ … +
z
n
+ ... or 
.
and 
the real and imaginary parts of the partial sum are indicated. A numerical sequence converges if and only if the sequences composed of its real and imaginary parts converge. Thus, a series with complex terms converges if and only if the series formed by its real and imaginary parts converge. One of the methods for studying the convergence of series with complex terms is based on this assertion.
.
: Further values are periodically repeated. A number of real parts: ; series of imaginary parts ; both series converge (conditionally), so the original series converges.19.4.1.2. Absolute convergence.
called absolutely convergent if the series converges 
, composed of the absolute values of its members.
, then the series necessarily converges 
(
, so the series formed by the real and imaginary parts of the series 
, converge absolutely). If the row 
converges, and the series 
diverges, then the series 
is called conditionally convergent.
is a series with non-negative terms, therefore, to study its convergence, all known signs (from comparison theorems to the Cauchy integral criterion) can be used.
.
. This series converges (the Cauchy test 
), so the original series converges absolutely.
.
, then any of its remainder converges. Conversely, if any remainder of the series converges, then the series itself converges.
.
.
and
, then their product for an arbitrary order of terms also converges absolutely, and its sum is equal to
.
