Complex numbers and series with complex terms. Convergent series of complex numbers Absolutely convergent series of complex numbers
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Standard methods, but reached a dead end with another example.
What is the difficulty and where can there be a snag? Let's put aside the soapy rope, calmly analyze the reasons and get acquainted with the practical methods of solution.
First and most important: in the overwhelming majority of cases, to study the convergence of a series, it is necessary to apply some familiar method, but the common term of the series is filled with such tricky stuffing that it is not at all obvious what to do with it. And you go around in circles: the first sign does not work, the second does not work, the third, fourth, fifth method does not work, then the drafts are thrown aside and everything starts anew. This is usually due to a lack of experience or gaps in other sections of calculus. In particular, if running sequence limits and superficially disassembled function limits, then it will be difficult.
In other words, a person simply does not see the necessary solution due to a lack of knowledge or experience.
Sometimes “eclipse” is also to blame, when, for example, the necessary criterion for the convergence of the series is simply not fulfilled, but due to ignorance, inattention or negligence, this falls out of sight. And it turns out like in that bike where the professor of mathematics solved a children's problem with the help of wild recurrent sequences and number series =)
In the best traditions, immediately living examples: rows and their relatives - diverge, since in theory it is proved sequence limits. Most likely, in the first semester, you will be beaten out of your soul for a proof of 1-2-3 pages, but now it is quite enough to show that the necessary condition for the convergence of the series is not met, referring to known facts. Famous? If the student does not know that the root of the nth degree is an extremely powerful thing, then, say, the series
put him in a rut. Although the solution is like two and two: , i.e. for obvious reasons, both series diverge. A modest comment “these limits have been proven in theory” (or even its absence at all) is quite enough for offset, after all, the calculations are quite heavy and they definitely do not belong to the section of numerical series.
And after studying the next examples, you will only be surprised at the brevity and transparency of many solutions:
Example 1
Investigate the convergence of a series
Solution: first of all, check the execution necessary criterion for convergence. This is not a formality, but a great chance to deal with the example of "little bloodshed".
"Inspection of the scene" suggests a divergent series (the case of a generalized harmonic series), but again the question arises, how to take into account the logarithm in the numerator?
Approximate examples of tasks at the end of the lesson.
It is not uncommon when you have to carry out a two-way (or even three-way) reasoning:
Example 6
Investigate the convergence of a series
Solution: first, carefully deal with the gibberish of the numerator. The sequence is limited: . Then:
Let's compare our series with the series . By virtue of the double inequality just obtained, for all "en" it will be true:
Now let's compare the series with the divergent harmonic series.
Fraction denominator less the denominator of the fraction, so the fraction itself – more fractions (write down the first few terms, if not clear). Thus, for any "en":
So, by comparison, the series diverges along with the harmonic series.
If we change the denominator a little: , then the first part of the reasoning will be similar:
. But to prove the divergence of the series, only the limit test of comparison is already applicable, since the inequality is false.
The situation with converging series is “mirror”, that is, for example, for a series, both comparison criteria can be used (the inequality is true), and for a series, only the limiting criterion (the inequality is false).
We continue our safari through the wild, where a herd of graceful and succulent antelopes loomed on the horizon:
Example 7
Investigate the convergence of a series
Solution: the necessary convergence criterion is satisfied, and we again ask the classic question: what to do? Before us is something resembling a convergent series, however, there is no clear rule here - such associations are often deceptive.
Often, but not this time. By using Limit comparison criterion Let's compare our series with the convergent series . When calculating the limit, we use wonderful limit , where as infinitesimal stands:
converges together with next to .
Instead of using the standard artificial technique of multiplication and division by a "three", it was possible to initially compare with a convergent series.
But here a caveat is desirable that the constant-multiplier of the general term does not affect the convergence of the series. And just in this style the solution of the following example is designed:
Example 8
Investigate the convergence of a series
Sample at the end of the lesson.
Example 9
Investigate the convergence of a series
Solution: in the previous examples, we used the boundedness of the sine, but now this property is out of play. The denominator of a fraction of a higher order of growth than the numerator, so when the sine argument and the entire common term infinitely small. The necessary condition for convergence, as you understand, is satisfied, which does not allow us to shirk from work.
We will conduct reconnaissance: in accordance with remarkable equivalence , mentally discard the sine and get a series. Well, something like that….
Making a decision:
Let us compare the series under study with the divergent series . We use the limit comparison criterion:
Let us replace the infinitesimal with the equivalent one: for .
A finite number other than zero is obtained, which means that the series under study diverges along with the harmonic series.
Example 10
Investigate the convergence of a series
This is a do-it-yourself example.
For planning further actions in such examples, the mental rejection of the sine, arcsine, tangent, arctangent helps a lot. But remember, this possibility exists only when infinitesimal argument, not so long ago I came across a provocative series:
Example 11
Investigate the convergence of a series .
Solution: it is useless to use the limitedness of the arc tangent here, and the equivalence does not work either. The output is surprisingly simple:
Study Series diverges, since the necessary criterion for the convergence of the series is not satisfied.
The second reason"Gag on the job" consists in a decent sophistication of the common member, which causes difficulties of a technical nature. Roughly speaking, if the series discussed above belong to the category of “figures you guess”, then these ones belong to the category of “you decide”. Actually, this is called complexity in the "usual" sense. Not everyone will correctly resolve several factorials, degrees, roots and other inhabitants of the savannah. Of course, factorials cause the most problems:
Example 12
Investigate the convergence of a series
How to raise a factorial to a power? Easily. According to the rule of operations with powers, it is necessary to raise each factor of the product to a power:
And, of course, attention and once again attention, the d'Alembert sign itself works traditionally:
Thus, the series under study converges.
I remind you of a rational technique for eliminating uncertainty: when it is clear order of growth numerator and denominator - it is not at all necessary to suffer and open the brackets.
Example 13
Investigate the convergence of a series
The beast is very rare, but it is found, and it would be unfair to bypass it with a camera lens.
What is double exclamation point factorial? The factorial "winds" the product of positive even numbers:
Similarly, the factorial “winds up” the product of positive odd numbers:
Analyze what is the difference between
Example 14
Investigate the convergence of a series
And in this task, try not to get confused with the degrees, wonderful equivalences and wonderful limits.
Sample solutions and answers at the end of the lesson.
But the student gets to feed not only tigers - cunning leopards also track down their prey:
Example 15
Investigate the convergence of a series
Solution: the necessary criterion of convergence, the limiting criterion, the d'Alembert and Cauchy criteria disappear almost instantly. But worst of all, the feature with inequalities, which has repeatedly rescued us, is powerless. Indeed, comparison with a divergent series is impossible, since the inequality incorrect - the multiplier-logarithm only increases the denominator, reducing the fraction itself
in relation to the fraction. And another global question: why are we initially sure that our series
is bound to diverge and must be compared with some divergent series? Does he fit in at all?
Integral feature? Improper integral evokes a mournful mood. Now, if we had a row
… then yes. Stop! This is how ideas are born. We make a decision in two steps:
1) First, we study the convergence of the series . We use integral feature:
Integrand continuous on the
Thus, a number diverges together with the corresponding improper integral.
2) Compare our series with the divergent series . We use the limit comparison criterion:
A finite number other than zero is obtained, which means that the series under study diverges along with side by side .
And there is nothing unusual or creative in such a decision - that's how it should be decided!
I propose to independently draw up the following two-move:
Example 16
Investigate the convergence of a series
A student with some experience in most cases immediately sees whether the series converges or diverges, but it happens that a predator cleverly disguises itself in the bushes:
Example 17
Investigate the convergence of a series
Solution: at first glance, it is not at all clear how this series behaves. And if we have fog in front of us, then it is logical to start with a rough check of the necessary condition for the convergence of the series. In order to eliminate uncertainty, we use an unsinkable multiplication and division method by adjoint expression:
The necessary sign of convergence did not work, but brought our Tambov comrade to light. As a result of the performed transformations, an equivalent series was obtained , which in turn strongly resembles a convergent series .
We write a clean solution:
Compare this series with the convergent series . We use the limit comparison criterion:
Multiply and divide by the adjoint expression:
A finite number other than zero is obtained, which means that the series under study converges together with next to .
Perhaps some have a question, where did the wolves come from on our African safari? Don't know. They probably brought it. You will get the following trophy skin:
Example 18
Investigate the convergence of a series
An example solution at the end of the lesson
And, finally, one more thought that visits many students in despair: instead of whether to use a rarer criterion for the convergence of the series? Sign of Raabe, sign of Abel, sign of Gauss, sign of Dirichlet and other unknown animals. The idea is working, but in real examples it is implemented very rarely. Personally, in all the years of practice, I have only 2-3 times resorted to sign of Raabe when nothing really helped from the standard arsenal. I reproduce the course of my extreme quest in full:
Example 19
Investigate the convergence of a series
Solution: Without any doubt a sign of d'Alembert. In the course of calculations, I actively use the properties of degrees, as well as second wonderful limit:
Here's one for you. D'Alembert's sign did not give an answer, although nothing foreshadowed such an outcome.
After going through the manual, I found a little-known limit proven in theory and applied a stronger radical Cauchy criterion:
Here's two for you. And, most importantly, it is not at all clear whether the series converges or diverges (an extremely rare situation for me). Necessary sign of comparison? Without much hope - even if in an unthinkable way I figure out the order of growth of the numerator and denominator, this still does not guarantee a reward.
A complete d'Alembert, but the worst thing is that the series needs to be solved. Need. After all, this will be the first time that I give up. And then I remembered that there seemed to be some more powerful signs. Before me was no longer a wolf, not a leopard and not a tiger. It was a huge elephant waving a big trunk. I had to pick up a grenade launcher:
Sign of Raabe
Consider a positive number series.
If there is a limit , then:
a) At a row diverges. Moreover, the resulting value can be zero or negative.
b) At a row converges. In particular, the series converges for .
c) When Raabe's sign does not give an answer.
We compose the limit and carefully simplify the fraction:
Yes, the picture is, to put it mildly, unpleasant, but I was no longer surprised. lopital rules, and the first thought, as it turned out later, turned out to be correct. But first, for about an hour, I twisted and turned the limit using “usual” methods, but the uncertainty did not want to be eliminated. And walking in circles, as experience suggests, is a typical sign that the wrong way of solving has been chosen.
I had to turn to Russian folk wisdom: "If nothing helps, read the instructions." And when I opened the 2nd volume of Fichtenholtz, to my great joy I found a study of an identical series. And then the solution went according to the model.
1. Complex numbers. Complex numbers called numbers of the form x+iy, where X and y - real numbers, i-imaginary unit, defined by equality i 2 =-1. Real numbers X and at are called respectively valid and imaginary parts complex number z. For them, the notation is introduced: x=Rez; y=imz.
Geometrically, every complex number z=x+iy represented by a dot M (x; y) coordinate plane xOy(Fig. 26). In this case the plane hoy called the complex number plane, or the plane of the complex variable z.
Polar coordinates r and φ points M, which is the image of a complex number z, are called module and argument complex number z; the notation is introduced for them: r=|z|, φ=Argz.
Since each point of the plane corresponds to an infinite number of values of the polar angle, which differ from each other by 2kπ (k is a positive or negative integer), Arg is a z-infinite-valued function of z.
That of the values of the polar angle φ , which satisfies the inequality –π< φ ≤ π are called main importance argument z and denote arg z.
In the following, the designation φ save only for the main value of the argument z , those. let's put φ =argz, whereby for all other values of the argument z we get the equality
Arg z = arg z + 2kπ =φ + 2kπ.
The relations between the modulus and argument of the complex number z and its real and imaginary parts are established by the formulas
x = r cos φ; y = r sin φ.
Argument z can also be determined by the formula
arg z = arctg (y / x) + C,
where FROM= 0 at x > 0, FROM= +π for x<0, at> 0; C \u003d - π at x < 0, at< 0.
Replacing x and at in complex number notation z = x+iy their expressions through r and φ , we get the so-called trigonometric form of a complex number:
Complex numbers z 1 \u003d x 1 + iy 1 and z 2 \u003d x 2 + iy 2 considered equal if and only if their real and imaginary parts are equal separately:
z1 = z2, if x 1 = x 2, y 1 = y 2 .
For numbers given in trigonometric form, equality takes place if the modules of these numbers are equal, and the arguments differ by an integer multiple of 2π:
z 1 = z 2, if |z 1 | = |z 2 | and Arg z 1 = Arg z 2 +2kπ.
Two complex numbers z = x+iy and z = x -iy with equal real and opposite imaginary parts are called conjugated. For conjugate complex numbers, the relations
|z 1 | = |z 2 |; arg z 1 = -arg z 2,
(the last equality can be given the form Arg z 1 + Arg z 2 = 2kπ).
Operations on complex numbers are defined by the following rules.
Addition. If a z 1 \u003d x 1 + iy 1, z 2 \u003d x 2 + iy 2, then
The addition of complex numbers obeys the commutative and associative laws:
Subtraction. If a , then
For a geometric explanation of the addition and subtraction of complex numbers, it is useful to represent them not as points on the plane z, and vectors: the number z = x + iy represented by vector having the beginning at the point O ("zero" point of the plane - the origin of coordinates) and the end at the point M(x; y). Then the addition and subtraction of complex numbers is performed according to the rule of addition and subtraction of vectors (Fig. 27).
Such a geometric interpretation of the operations of addition and subtraction of vectors makes it easy to establish theorems on the modulus of the sum and difference of two and the sum of several complex numbers, expressed by the inequalities:
| |z 1 |-|z 2 | | ≤ |z 1 ±z 2 | ≤ |z 1 | + |z 2 | ,
In addition, it is useful to remember that modulus of the difference of two complex numbers z1 and z2 is equal to the distance between the points that are their images on the z plane:| |z 1 -z 2 |=d(z 1 ,z 2) .
Multiplication. If a z 1 \u003d x 1 + iy 1, z 2 \u003d x 2 + iy 2. then
z 1 z 2 \u003d (x 1 x 2 -y 1 y 2) + i (x 1 y 2 + x 2 y 1).
Thus, complex numbers are multiplied as binomials, with i 2 replaced by -1.
IF , then
In this way, the modulus of the product is equal to the product of the modules of the somnoektels, and the argument of the product-the sum of the arguments of the factors. The multiplication of complex numbers obeys the commutative, associative and distributive (with respect to addition) laws:
Division. To find the quotient of two complex numbers given in algebraic form, the dividend and the divisor should be multiplied by the number conjugate to the divisor:
" If a given in trigonometric form, then
In this way, the modulus of the quotient is equal to the quotient of the modulus of the dividend and divisor, a argument private is equal to the difference between the arguments of the dividend and the divisor.
Exponentiation. If z= , then by the Newton binomial formula we have
(P is a positive integer); in the resulting expression, it is necessary to replace the degrees i their meanings:
i 2 \u003d -1; i 3 =i; i 4 =1; i 5 =1,…
and, in general,
i 4k = 1; i 4k+1 =i; i 4k+2 = -1; i 4k+3 = -i .
If , then
(here P can be either a positive integer or a negative integer).
In particular,
(De Moivre's formula).
Root extraction. If a P is a positive integer, then the nth root of the complex number z has n different values, which are found by the formula
where k=0, 1, 2, ..., n-1.
437.
Find (z 1 z 2)/z 3 if z1 = 3 + 5i, z 2 = 2 + 3i, z 3 = 1+2i.
∆
438.
number z= 2 + 5i.
∆ Find the modulus of the complex number: . Find the main value of the argument: . Therefore, ▲
439.
Represent in trigonometric form the complex
number
∆ Find , ; , , i.e.
440.
Represent in trigonometric form complex
numbers 1, i, -1, -i.
441.
Represent Numbers ,
,
in trigonometric form and then find the complex number
z 1 /(z 2 z 3).
∆ Find
Consequently,
442. Find all values.
∆ We write the complex number in trigonometric form. We have , , . Consequently,
Consequently, , ,
443. Solve a binary equation ω 5 + 32i = 0.
∆ Let us rewrite the equation in the form ω 5 + 32i = 0. Number -32i represent in trigonometric form:
If a k = 0 then (A).
k=1,(B).
k=2,(C).
k=3,(D).
k=4,(E).
The roots of the two-term equation correspond to the vertices of a regular pentagon inscribed in a circle of radius R=2 centered at the origin (Fig. 28).
In general, the roots of a two-term equation ω n \u003d a, where a-complex number, correspond to the vertices of the regular n-gon inscribed in a circle with center at the origin and radius equal to ▲
444. Using De Moivre's formula, express cos5φ and sin5 φ through cosφ and sinφ.
∆ We transform the left side of the equality according to the Newton binomial formula:
It remains to equate the real and imaginary parts of the equality:
445. Given a complex number z=2-2i. Find Rez, Imz, |z|, argz.
446. z = -12 + 5i.
447 . Calculate the expression using the Moivre formula (cos 2° + isin 2°) 45 .
448. Calculate using De Moivre's formula.
449. Express a complex number in trigonometric form
z = 1 + cos 20° + isin 20°.
450. Evaluate expression (2 + 3i) 3 .
451.
Evaluate expression
452. Evaluate expression
453. Express a complex number in trigonometric form 5-3i.
454. Express a complex number in trigonometric form -1 + i.
455.
Evaluate expression
456.
Evaluate expression having previously presented the factors in the numerator and denominator in trigonometric form.
457. Find all values
458.
Solve a binary equation
459. express cos4φ and sin4φ through cosφ and sinφ.
460. Show that the distance between points z1 and z2 equals | z2-z1|.
∆ We have z 1 \u003d x 1 + iy 1, z 2 \u003d x 2 + iy 2, z 2 -z 1 \u003d (x 2 -x 1) + i (y 2 -y 1), where
those. | z2-z1| is equal to the distance between the given points. ▲
461. Which line is described by the point z, satisfying the equation where With-constant complex number, and R>0?
462.
What is the geometric meaning of the inequalities: 1) | z-c|
463. What is the geometric meaning of the inequalities: 1) Rez > 0; 2) im z< 0 ?
2. Series with complex terms. Consider the sequence of complex numbers z 1 , z 2 , z 3 , ..., where z p \u003d x p + iy p (n \u003d 1, 2, 3, ...). constant number c = a + bi called limit sequences z 1 , z 2 , z 3 , ..., if for any arbitrarily small number δ>0 there is a number N, what is the meaning z p with numbers n > N satisfy the inequality \z n-With\< δ . In this case, write .
A necessary and sufficient condition for the existence of a limit of a sequence of complex numbers is as follows: the number c=a+bi is the limit of the sequence of complex numbers x 1 + iy 1, x 2 + iy 2, x 3 + iy 3, ... if and only if , .
(1)
whose members are complex numbers is called converging, if nth partial sum of the series S n for n → ∞ tends to a certain end limit. Otherwise, series (1) is called divergent.
Series (1) converges if and only if series with real terms converge
(2) Investigate the convergence of the series This series, whose terms form an infinitely decreasing geometric progression, converges; therefore, the given series with complex terms converges absolutely. ^
474. Find the area of convergence of a series
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 − diverges (due to the imaginary part).
If the real and imaginary parts of a series converge absolutely, then the
row, because . The converse is also true: from the absolute convergence of the complex series
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 . Moreover, in any closed 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 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.
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: Substitute in a series of modules
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. . This is the basis for determining exponential function in the complex area:
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: The rest of the relations are obtained similarly. So:
Examples. Represent these expressions in the form
2. (the expression in brackets is a number i
, written in exponential form)
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 region.
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 we have:
, which proves the theorem.
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.
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 criterion.
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. Exponential and trigonometric functions of a 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 a sequence, the sum of a series are formed and proved in exactly 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 a 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 a
for anyone z owned D, the following inequality holds:
then the row (1) is called uniformly convergent in the region D.
Series with complex terms.
19.3.1. 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.
19.3.1.1. Basic definitions. Let an infinite sequence of complex numbers be given. The real part of the number will be denoted by , the imaginary - (i.e. .
Number series- view record .
Partial sums of a series:
Definition. If there is a limit S sequences of partial sums of the series with , which is a proper complex number, then the series is said to converge; number S called the sum of the series and write or .
Find the real and imaginary parts of the partial sums: , where the symbols and denote the real and imaginary parts of the partial sum. 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.
Example.
19.3.1.2. Absolute convergence.
Definition. The row is called absolutely convergent if the series converges , composed of the absolute values of its members.
Just as for numerical real series with arbitrary terms, it can be proved that if the series converges, then the series necessarily converges. If the series converges and the series diverges, then the series is said to be conditionally convergent.
A series is a series with non-negative members, therefore, to study its convergence, all known features can be used (from comparison theorems to the Cauchy integral test).
Example. Investigate the series for convergence.
Let's make a series of modules (): . This series converges (the Cauchy test ), so the original series converges absolutely.
19.1.3.4. 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, then any of its remainder converges. Conversely, if any remainder of the series converges, then the series itself 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 number With, then the convergence of the series is preserved, and the sum is multiplied by With.
Convergent rows ( BUT) and ( AT) 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 ( BUT) and ( AT) converge absolutely to their sumand, then their product for an arbitrary order of terms also converges absolutely, and its sum is equal to.
19.3.2. Power complex series.
Definition. A power series with complex terms is a series of the form
where are constant complex numbers (coefficients of the series), is a fixed complex number (the center of the circle of convergence). For any numerical value z the series turns into a numerical series with complex terms, converging or diverging. If the series converges at a point z , then this point is called the convergence point of the series. The power series has at least one point of convergence - the point . The set of points of convergence is called the region of convergence of the series.
As for a power series with real terms, all meaningful information about a power series is contained in Abel's theorem.
Abel's theorem. If the power series converges at the point , then
1. it absolutely converges at any point on the circle ;
2. If this series diverges at , then it diverges at any point z
, satisfying the inequality (i.e., located farther from the point than ).
The proof repeats verbatim the proof of the section 18.2.4.2. Abel's theorem for a series with real members.
Abel's theorem implies the existence of such a non-negative real number R , that the series converges absolutely at any interior point of a circle of radius R centered at , and diverges at any point outside this circle. Number R called radius of convergence, a circle - circle of convergence. At the points of the boundary of this circle - circles of radius R centered at a point - the series can both converge and diverge. At these points, the series of modules has the form . The following cases are possible:
1. The series converges. In this case, the series converges absolutely at any point on the circle.
2. The series diverges, but its common term . In this case, the series can conditionally converge at some points of the circle, and diverge at others, i.e. each point requires an individual study.
3. The series diverges, and its common term does not tend to zero at . In this case, the series diverges at any point of the boundary circle.