Series with complex terms. Series in the complex domain Number series with complex numbers

463. What is the geometric meaning of the inequalities: 1) Re z > 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 = x p + iу p (p = 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 such a number N, what is the meaning z p with numbers n > N satisfy the inequality \z p-With\< δ . In this case they 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 a sequence of complex numbers x 1 +iу 1, x 2 +iу 2, x 3 +iу 3, … if and only if , .

(1)

whose members are complex numbers is called convergent, If nth partial sum of the series S n at p → ∞ tends to a certain final 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, the terms of which form an infinitely decreasing geometric progression, converges; therefore, a given series with complex terms converges absolutely. ^

474. Find the area of ​​convergence of the series

Transcript

1 Federal Agency for Education Tomsk State University of Architecture and Civil Engineering ROWS WITH COMPLEX MEMBERS Guidelines for independent work Compiled by LI Lesnyak, VA Starenchenko Tomsk

2 Rows with complex members: methodological instructions / Compiled by LI Lesnyak, VA Starenchenko - Tomsk: Tomsk State Architectural and Construction University Publishing House, with Reviewer Professor NN Belov Editor EY Glotova Methodical instructions are intended for self-study by 1st year students of all specialties topics “Series with complex members” of the JNF discipline “Mathematics” Published according to the decision of the methodological seminar of the department of higher mathematics, protocol 4 of March Approved and put into effect by the vice-rector for academic affairs VV Dzyubo from 5 to 55 The original layout was prepared by the author Signed for printing Format 6 84/6 Offset paper Typeface Times Educational publication l, 6 Circulation 4 Order Publishing house TGASU, 64, Tomsk, Solyanaya sq., Printed from the original layout in the OOP TGASU 64, Tomsk, Partizanskaya st., 5

3 SERIES WITH COMPLEX TERMS TOPIC Number series with complex terms Recall that complex numbers are numbers of the form z = x y, where x and y are real numbers, and the imaginary unit defined by the equality = - The numbers x and y are called the real and imaginary parts of the number z, respectively and denote x = Rez, y = Imz Obviously, between the points M(x, y) of the XOU plane with a Cartesian orthogonal coordinate system and complex numbers of the form z = x y, there is a one-to-one correspondence. The XOU plane is called the complex plane, and z is called a point of this plane Real numbers correspond to the abscissa axis, called the real axis, and numbers of the form z = y correspond to the ordinate axis, which is called the imaginary axis. If the polar coordinates of the point M(x,y) are denoted by r and j, then x = r cosj, y = r s j and the number z will be written in the form: z = r (cosj sj), where r = x y This form of writing a complex number is called trigonometric, writing z in the form z = x y is called an algebraic form of writing The number r is called the modulus of the number z, the number j is the argument (at the point z = the concept of an argument is not extended) The modulus of the number z is uniquely determined by the formula z = x y The argument j is uniquely determined only under the additional condition - π< j π (или j < π), обозначается в этом случае arq z и называется главным значением аргумента

4 numbers z (fig) The meaning of this should be remembered that y arq z - π is expressed through< arctg y x < π y arctg, при x r = z = x y М (x, y) j = arq z Рис x Если считать, что - π < arg z π, то y arg z = arctg, если х >,y; x y arg z = -arctg, if x >, y< ; х у arg z = -π arctg, если х <, y < ; х у arg z = π - arctg, если х <, y ; х π arg z =, если х =, y >; π arg z = -, if x =, y< Например, если z = - (х <, y >), 4

5 π arg z = π - arctg = π - = π ; z = = (fig) М y r = j = p x Fig In trigonometric form, the number z = - will be written in the form: - = сos π s π и It is recommended to repeat operations on complex numbers yourself. Let us only recall the formula for raising the number z to a power: z = ( x y) = r (cosj s j) 5

6 6 Key questions of the theory Brief answers Definition of a series with complex terms The concept of convergence of a series Necessary condition for convergence Definition Let a sequence z ) = ( x y ) = z, z, z, of complex numbers be given A symbol of the form ( å = z is called a series, z is a general term of the series The concepts of partial sums of a series S, its convergence and divergence fully correspond to similar concepts for series with real terms. The sequence of partial sums of a series has the form: S = z ; S = z z ; S = z z z ; If $lm S and this limit is finite and equal to the number S , the series is called convergent, and the number S is called the sum of the series, otherwise the series is called divergent. Recall that the definition of the limit of a sequence of complex numbers, which we used, is formally no different from the definition of the limit of a sequence of real numbers: def (lm S = S) = (" ε > $ N > : > N Þ S - S< ε) Как и в случае рядов с действительными членами, необходимым условием сходимости ряда å = z является стремление к

7 zero of the general term z of the series at This means that if this condition is violated, that is, if lm z ¹, the series diverges, but if lm z =, the question of the convergence of the series remains open. Is it possible to study the series å (x = for convergence by investigating x and å = for the convergence of series å = with real terms? y y) Yes, it is possible. The following theorem holds: Theorem In order for the series å = y (x) to converge, it is necessary and sufficient that both series å = å = converge. y, and if å x = S = where å S = (x y) = å = x u, and y = S, then S = S S, converges - Example Make sure that the series å = è () xia, and find its sum is 7

8 Solution The series å converges, t k ~ = () () when The sum S of this series is equal to (Chapter, topic, n) The series å converges as an infinitely decreasing geometric = progression, with å = () и S b = - q = converges, and its sum Thus, the series S = Example Series å diverges, t k diverges = è! harmonic series å In this case, examine the series å = for convergence! does not make sense Example The series å π tg diverges, because for = è the series å π tg the necessary condition for convergence is violated = π lm tg = p ¹ и 8

9 What properties do convergent series with complex terms have? The properties are the same as those of convergent series with real terms. It is recommended to repeat the properties. 4 Is there a concept of absolute convergence for a series with complex terms? Theorem (sufficient condition for the convergence of a series) If the series å = z converges, then the series å = z will also converge. The concept of absolute convergence of the series å = z formally looks exactly the same as for series with real terms. Definition The series å = z is called absolutely convergent, if the series converges å = z Example Prove the absolute convergence of the series () () () 4 8 Solution Let’s use the trigonometric form of writing the number: 9

10 π π = r (cosj s j) = cos s и 4 4 Then π π () = () cos s Þ и 4 4 () π π Þ = cos s Þ z = 4 4 и It remains to examine the series å z for convergence = = This is an infinitely decreasing geometric progression with a denominator; such a progression converges, and, therefore, the series converges absolutely. When proving absolute convergence, the theorem is often used. Theorem For the series å = y (x) to converge absolutely, it is necessary and sufficient that both series å = be absolutely Example Series å = (-) è cosπ ! x and å = y converges absolutely, t k converges absolutely å (-), and the absolute convergence = of the series å cosπ is easily proven: =!

11 cosπ, and the row is å!! =! converges by d'Alembert's criterion By the comparison criterion the series å cosπ converges Þ series å =! converges absolutely cosπ =! Solving problems Examine series 4 for convergence: å ; å (-) = è l l = è! l å = π - cos и α tan π ; 4 å = и и ;! Solution å = è l l The series diverges, because the series å diverges, which is easily established by the comparison test: >, and the harmonic = l l series å, as is known, diverges. Note that with = in this case the series å based on the integral Cauchy test = l converges å (-) = è! l

12 The series converges, so to å =! converges on the basis of d'Alembert's limit test, and the series å (-) converges according to the theorem = l Leibniz å α π - π cos tg = и и Obviously, the behavior of the series will depend on the exponent α Let us write the series using the formula β - cosβ = s: å α π π s tg = и At α< ряд будет расходиться, т к α π lm s ¹ Þ ряд å π s расходится, а это будет означать, что расходится и данный è = è ряд α π α π cost При α >s ~ = Series α å и и 4 = will converge provided that α >, i.e. for α > and will diverge for α or for will converge, since for π π tg ~ α Series å = α α π tg α

13 Thus, the original series will converge at and diverge at α 4 å = и и! α > The series å is examined for convergence using = è Cauchy's limit test: lm = lm = > Þ è the series diverges Þ e è Þ will diverge and the original series 5 series Series 5 6 is examined for absolute convergence π cos ; 6 å (8) (-)! =! å = Solution 5 å = π cos()! å = - π cos converges absolutely, so to (-)! converges according to the comparison criterion: π cos, and the series å (-)! (-)! = (-)! converges according to d'Alembert's test

14 4 6 å =!) 8 (To the row!) 8 (å = apply d'Alembert's sign:!) 8 (:)! () 8 (lm = 8 8 lm = 8 lm = = Þ< = lm ряд сходится Это означает, что данный ряд сходится абсолютно Банк задач для самостоятельной работы Ряды 6 исследовать на сходимость å = è ; å = è π s! 5 ; å = è π s! 5 ; 4 å = è è - l) (; 5 å = - è π tg e ; 6 å = è l Ответы:, 6 расходятся;, 4, 5 сходятся

15 5 Examine series 7 for absolute convergence 7 å = è - π s) (; 8! å = è ; 9 å = è - 5 π s) (; å = è -! 5) (Answers: 7, 8 converge absolutely , 9 diverges, does not converge absolutely

16 TOPIC Power series with complex terms When studying the section “Functional series”, series were considered in detail, the terms of which were members of a certain sequence of functions of a real variable. The most attractive (especially in terms of applications) were power series, i.e. series of the form å = a (x-x) It was proven (Abel's theorem) that every power series has an interval of convergence (x - R, x R), within which the sum S (x) of the series is continuous and that the power series within the convergence interval can be differentiated term by term and integrated term by term. These are the remarkable properties of power series have opened up the broadest possibilities for their numerous applications. In this topic we will consider power series not with real, but with complex terms 6 Key questions of theory Short answers Definition of a power series A power series is a functional series of the form å = a (z - z), () where a and z are given complex numbers, and z is a complex variable. In the special case when z =, the power series has the form å = a z ()

17 Obviously, the series () is reduced to the series () by introducing a new variable W = z - z, so we will mainly deal with series of the form () Abel’s theorem If the power series () converges at z = z ¹, then it converges and, moreover, absolutely for any z for which z< z Заметим, что и формулировка, и доказательство теоремы Абеля для рассмотренных ранее степенных рядов å aх ничем = не отличается от приведенной теоремы, но геометрическая иллюстрация теоремы Абеля разная Ряд å = условия х a х при выполнении х < будет сходиться на интервале - х, х) (рис), y (а для ряда с комплексными членами условие z z < означает, что ряд будет сходиться внутри круга радиуса z (рис 4) x x x - x z z x Рис Рис 4 7

18 Abel’s theorem has a corollary, which states that if the series å = a z diverges for * z = z, then it will also diverge for any z for which * z > z Is there a concept of radius for power series () and () convergence? Yes, there is a Radius of Convergence R, a number that has the property that for all z, for which z< R, ряд () сходится, а при всех z, для которых z >R, series () diverges 4 What is the region of convergence of series ()? If R is the radius of convergence of the series (), then the set of points z for which z< R принадлежит кругу радиуса R, этот круг называют кругом сходимости ряда () Координаты точек М (х, у), соответствующих числам z = x y, попавшим в круг сходимости, будут удовлетворять неравенству x < y R Очевидно, круг сходимости ряда å a (z - z) имеет центр = уже не в начале координат, а в точке М (х, у), соответствующей числу z Координаты точек М (х, у), попавших в круг сходимости, будут удовлетворять неравенству (x - х) (y - у < R) 8

19 5 Is it possible to find the radius of convergence a using the formulas R = lm and R = lm, a a which took place for power series with real terms? It is possible, if these limits exist If it turns out that R =, this will mean that the series () converges only at the point z = or z = z for the series () When R = the series will converge on the entire complex plane Example Find the radius of convergence of the series å z = a Solution R = lm = lm = a Thus, the series converges inside a circle of radius. The example is interesting because on the boundary of the circle x y< есть точки, в которых ряд сходится, и есть точки, в которых расходится Например, при z = будем иметь гармонический ряд å, который расходится, а при = z = - будем иметь ряд å (-), который сходится по теореме = Лейбница Пример Найти область сходимости ряда å z =! Решение! R = lm = lm () = Þ ряд сходится ()! на всей комплексной плоскости 9

20 Recall that the power series å = a x within their convergence interval converge not only absolutely, but also uniformly. A similar statement holds for the series å = a z: if a power series converges and the radius of its convergence is equal to R, then this series in any closed circle z r provided that r< R, будет сходиться абсолютно и равномерно Сумма S (z) степенного ряда с комплексными членами внутри круга сходимости обладает теми же свойствами, что и сумма S (x) степенного ряда å a х внутри интервала сходимости Свойства, о которых идет речь, рекомендуется = повторить 6 Ряд Тейлора функции комплексного переменного При изучении вопроса о разложении в степенной ряд функции f (x) действительного переменного было доказано, что если функция f (x) на интервале сходимости степенного ряда å a х представима в виде å f (x) = a х, то этот степенной ряд является ее рядом Тейлора, т е коэффициенты вычис- = = () f () ляются по формуле a =! Аналогичное утверждение имеет место и для функции f (z): если f (z) представима в виде f (z) = a a z a z

21 in a circle of radius R > convergence of the series, then this series is the Taylor series of the function f (z), i.e. f () f () f å = () (z) = f () z z = z!!! Coefficients of the series å = () f (z) a =! f () a (z - z) are calculated by the formula Recall that the definition of the derivative f (z) is formally given in exactly the same way as for the function f (x) of a real variable, i.e. f (z) = lm def f (z D z) - f (z) D z Dz The rules for differentiating the function f (z) are the same as the rules for differentiating the function of a real variable 7 In what case is the function f (z) called analytic at the point z? The concept of a function analytic at a point z is given by analogy with the concept of a function f (x) that is real analytic at a point x. Definition A function f (z) is called analytic at a point z if there exists R > such that in the circle z z< R эта функция представима степенным рядом, т е å = f (z) = a (z - z), z - z < R -

22 We emphasize once again that the representation of a function f (z) analytic at a point z in the form of a power series is unique, and this series is its Taylor series, that is, the coefficients of the series are calculated by the formula () f (z) a =! 8 Basic elementary functions of a complex variable In the theory of power series of functions of a real variable, the series expansion of the function e x was obtained: = å x x e, xî(-,) =! When solving the example of point 5, we were convinced that the series å z converges on the entire complex plane. In the special case for z = x, its sum is equal to e x This fact underlies the following - =! following idea: for complex values ​​of z, the function е z by definition is considered the sum of the series å z Thus, =! z e () def å z = =! Definition of functions ch z and sh z x - x Since ch = = å k e e x x, x О (-,) k = (k)! x - x e - e sh = = å x k = k x, (k)! x О (-,),

23 and the function e z is now defined for all complex z, then it is natural to take ch z = on the entire complex plane, def z - z e e def z - z e - e sh z = Thus: z -z k e - e z sh z = = hyperbolic sine ; (k)! å k = z - z å k e e z cosh z = = hyperbolic cosine; k = (k)! shz th z = hyperbolic tangent; chz chz cth z = hyperbolic cotangent shz Definition of functions s z and cos z Let us use the expansions obtained earlier: å k k (-) s x x = k = (k)!, å k k (-) x cos x =, k = (k)! series converge on the entire number line When replacing x in these series with z, we obtain power series with complex terms, which, as is easy to show, converge on the entire complex plane. This allows us to determine for any complex z the functions s z and cos z: å k k (- ) s z z = k = (k)! ; å k k (-) z cos z = (5) k = (k)!

24 9 Relationship between the exponential function and trigonometric functions in the complex plane Replacing in the series å z z e = =! z by z, and then by z, we get: =å z z e, å -z (-) z e = =! =! Since e ()) e k k = (-, we will have: z -z = å k = k (-) z (k)! k = cos z z - z k k e - e (-) z = å = s z k= (k) ! Thus: z -z z -z e e e - e сos z = ; s z = (6) From the obtained formulas follows another remarkable formula: z сos z s z = e (7) Formulas (6) and (7) are called Euler’s formulas. that these formulas are also valid for real z. In the special case for z = j, where j is a real number, formula (7) will take the form: j cos j sj = e (8) Then the complex number z = r (cos j s j) will be written in the form : j z = re (9) Formula (9) is called the exponential form of writing the complex number z 4

25 Formulas connecting trigonometric and hyperbolic functions The following formulas are easily proven: s z = sh z, sh z = s z, cos z = ch z, cos z = cos z Let’s prove the first and fourth formulas (it is recommended to prove the second and third yourself) Let’s use the formulas ( 6) Euler: - z z z - z s e - e e - e z = = = sh z ; z -z e e ch z = = cos z Using the formulas sh z = s z and ch z = cos z, it is easy to prove, at first glance, a surprising property of the functions s z and cos z. Unlike the functions y = s x and y = cos x, the functions s z and cos z are not limited in absolute value. In fact, if in the indicated formulas, in particular, z = y, then s y = sh y, cos y = ch y This means that on the imaginary axis s z and cos z are not limited in absolute value It is interesting that for s z and cos z all the formulas are valid, similar to the formulas for the trigonometric functions s x and cos x. The given formulas are quite often used when studying series for convergence Example Prove the absolute convergence of the series å s = Solution We examine the series å for convergence s = As was noted, the function s z bounded on the imaginary axis is not 5

26 is, therefore, we cannot use the comparison criterion. We will use the formula s = sh. Then å = å s sh = = We study the series å sh = using D'Alembert's criterion: - () - - sh () e - e e (e- e) e lm = lm =lm=< - - sh e - e e (- e) Таким образом, ряд å s = сходится Þ данный ряд сходится абсолютно Решение задач Число z = представить в тригонометрической и комплексной формах y π Решение r = =, tg j = = Þ j =, x 6 π 6 π π = cos s = e è 6 6 Найти область сходимости ряда å (8 -) (z) = Решение Составим ряд из абсолютных величин заданного ряда и найдем его радиус сходимости: a 8 - () () R = lm = lm = lm a =, 6

27 () since lm =, from the modules converges under the condition 8 - = 8 = Thus, the series z< Данный ряд при этом же условии сходится, т е внутри круга радиуса с центром в точке при z >points of the circle z = -, will converge, and outside this circle, that is, the series diverges. We study the behavior of the series at z =, the equation of which in the Cartesian coordinate system has the form x (y) = At ​​z = 9, the series of absolute values ​​will have the form : å 8 - = å = = that this series in a closed circle The resulting series converges, this means z converges absolutely Prove that the function å z z e = is periodic with period π (this property of the function e z significantly distinguishes it =! from the function e x) Proof We use the definition of a periodic function and formula (6) We need to make sure that z z e π = e, where z = x y Let us show that this is so: z π x y π x (y π) x (y e = e = e = e e x = e (cos(y π) s (y π)) = e So, e z is a periodic function!) x π = (cos y s y) = e x y = e z 7

28 4 Get a formula that connects the numbers e and π Solution Let's use the exponential form of writing j complex number: z = re For z = - we will have r =, j = π and, thus, π e = - () Amazing formula and this despite the fact that the appearance in mathematics of each of the numbers π, e and has nothing to do with the appearance of the other two! Formula () is also interesting because it turns out that the exponential function e z, unlike the function e x, can take negative values ​​e x 5 Find the sum of the series å cos x =! Solution Let's transform the series x x сos x s x e (e) å = å = å!! x (e) cos x = = s x e e = = =! cos x s x cos x = e e = e (cos(s x) s (s x)) Þ å = = cosx =! cos = e x cos(s x) When solving, we used the formula = cos x s x twice and the series expansion of the function (e x) e 6 Expand the function f (x) = e x cos x into a power series, using the series expansion of the function x() x x x x e = e e = e cos x e s x Solution x() x() x e = å = å!! = = π cos s и 4 π = 4 8

29 = å x π π () cos s =! и 4 4 Т к å x x() x x π e cos x = Ree Þ e cos x = () cos =! 4 The resulting series converges on the entire numerical axis, so to x π (x) () cos, and the series å (x)! 4! =! x< (докажите по признаку Даламбера) сходится при Банк задач для самостоятельной работы Представить в тригонометрической и показательной формах числа z =, z = -, z = -, z = 4 Построить в декартовой системе координат точки, соответствующие заданным числам Записать в алгебраической и тригонометрической формах числа e π и Используя формулу z = r (cosj s j), вычислить () и (e π) 4 Исследовать на сходимость ряд å e = Ответ Ряд сходится абсолютно 5 Исследовать ряд å z на сходимость в точках = z = и z = 4 Ответ В точке z ряд сходится абсолютно, в точке z ряд расходится 9

30 6 Find the radius R and the circle of convergence of the series 4 Investigate the behavior of the series at the boundary points of the circle of convergence (at points lying on the circle) å!(z -) ; å(z); = = å () z = () ; 4 å z = 9 Answers:) R =, series converges at point z = - ;) R =, series converges absolutely in a closed circle z with center at point z = - or subject to x (y) ;) R =, series converges absolutely in a closed circle z or subject to x y ; 4) R =, the series converges absolutely in a closed circle z or under the condition x y 9 7 Expand the function f (x) = e x s x, () x into a power series using the series expansion of the function e 8 Make sure that for any complex z will take place formulas: s z = s z cos z, s z cos z =, s (z π) = s z (use Euler’s formulas)

31 LIST OF RECOMMENDED READING Basic literature Piskunov, NS Differential and integral calculus for colleges / NS Piskunov T M: Nauka, 8 S 86 9 Fichtengolts, GM Fundamentals of mathematical analysis / GM Fichtengolts T - St. Petersburg: Lan, 9 48 s Vorobyov, NN Theory rows / NN Vorobyov - St. Petersburg: Lan, 8 48 s 4 Written, DT Lecture notes on higher mathematics Ch / DT Written M: Iris-press, 8 5 Higher mathematics in exercises and problems Ch / PE Danko, AG Popov, TY Kozhevnikova [ etc.] M: ONICS, 8 C Additional literature Kudryavtsev, LD Course of mathematical analysis / LD Kudryavtsev TM: Higher school, 98 C Khabibullin, MV Complex numbers: guidelines / MV Khabibullin Tomsk, TGASU, 9 6 s Moldovanova, EA Rows and complex analysis: textbook / EA Moldovanova, AN Kharlamova, VA Kilin Tomsk: TPU, 9

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(functional series power series domain of convergence order of finding the interval of convergence - example radius of the interval of convergence examples) Let an infinite sequence of functions be given, Functional

Series Number series General concepts Definition If each natural number is associated with a certain number according to a certain law, then the set of numbered numbers is called a number sequence,

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