Power Series

12 Key excerpts on "Power Series"

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  A Course of Mathematical Analysis

  International Series of Monographs on Pure and Applied Mathematics

  • A. F. Bermant, I. N. Sneddon, S. Ulam, M. Stark(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  SERIES 455 It may be remarked, in addition, that the applicability of the arithmetic operations to infinite series is guaranteed by the property of absolute convergence of the series. 3. Power Series 129. Taylor's series. Definition. A Power Series is a functional series of the form «o + «i(* — *o) + «2<* — *o) 2 H l· «n(* — *o) n H , each term being the product of a constant and a power (with positive integral exponent) of the difference x — x 0 (in particular, if x Q = 0 — of the independent variable x itself). The constants a 0 , a x , a 2 , . . ., a n , . . . are called the coefficients of the Power Series. We naturally arrive at Power Series when we use Taylor's formula to form approximations to functions by polynomials (Sec. 79) with indefinitely increasing accuracy (i.e. with δ η -> 0). If f(x) has derivatives up to and including order (n ~f 1) in a neigh-bourhood of x 0 , as we know, f(x) = f(x 0 ) + f'(x 0 ) (x - «b) + 27/(*ο) (x - *o) 2 + · · · • ' ' + ^ τ / ( Λ ) (xo) (* ~ *o) n + { n l l ) { f (n+1) (f) (* ~ *οΓ + where ξ is an intermediate point between x 0 and x, or /(*) = tf n (a-a 0 ) + R n (*) (see Sec. 78). For a given w, the approximate expression for f(x) as a polynomial iV^ (a; — x 0 ) : /(a;) ^ N n (x - x 0 ) is generally speaking the more accurate, the smaller the length of the interval in which the approximation is considered (since the error δ η will be less). Now let the interval [a, 6], in which the approx-imation is considered, remain unchanged. Then an increase in accuracy can often be achieved by increasing the order n of the Taylor for-mula. For, it is clear from the expression for the maximum error (see Sec. 79): _ M n + 1 ° n ~ (*+l)I (Ö a) 30* 456 COURSE OF MATHEMATICAL ANALYSIS where M n+1 ^ |/ (n+1) (a;) |, that the denominator increases rapidly with increase of n, and δ η can be expected to tend to zero.

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  Mathematical Analysis and Proof

  • David S G Stirling(Author)
  • 2009(Publication Date)
  • Woodhead Publishing

    (Publisher)

  12

  Functions Defined by Power Series

  12.1 Introduction

  We are led naturally to the study of those functions which can be represented as the sums of Power Series for two reasons. Taylor’s Theorem allows us to express a function f as a sum of finitely many terms of the form (x – a )

  n f

  (n )

  (a )/n ! plus a remainder, and we may observe that in some cases the remainder tends to zero as the number of terms increases; this may be thought of as expressing f as a sum of the simpler functions (x – a )n . More significantly, we arrive at Power Series by attempting to find functions with desirable properties, usually arising from differential equations. If we wish to find a function equal to its own derivative, then if this function were expressible in the form ∑∞

  n =0

  a n x n , and if the derivative turned out to equal the expression obtained by differentiating term by term, ∑∞

  n =1

  na n x

  n −1

  , then the equality of the function and its derivative would be guaranteed if we were to choose the coefficients

  an

  so that these two series were identical, that is,

  an  = a

  0 /n !. Assuming this outline programme is correct and, in particular, that the interchange of the limits involved in taking derivatives of infinite sums can be justified, this yields a simple technique for producing various special functions required in everyday mathematics.

  In what follows we shall consider functions of the form ∑∞

  n =0

  a n x n where

  an

  does not depend on x and the term a 0 x 0 is understood to mean a 0 . By a simple change of variable this allows us to consider functions of the form ∑ a n (x  − a )n .

  The first thing to recall is the result of Lemma 8.11, that if the Power Series ∑ a n w n converges, then ∑ a n x n converges if |x | < |w

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  A Student's Guide to Infinite Series and Sequences

  • Bernhard W. Bach, Jr.(Authors)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  Many mathematical functions have a Power Series representation or expansion . If a function has a Power Series expansion (not all do), then it is unique, and the Power Series will converge to that function (within the interval of convergence). Functions de fi ned in this manner have properties similar to those of polynomials and can be treated as such. That is, they can be added, subtracted, multiplied, differentiated, and so forth. The usefulness of Power Series representations lies in the fact that the series representation gives us a simple technique for evaluating functions, their integrals, and their deriva-tives. For example, if the function f ( x ) has the Power Series expansion f ð x Þ ¼ a 0 þ a 1 x þ a 2 x 2 þ a 3 x 3 þ … ; then the integral of f ( x ) can be found by integrating the Power Series term by term: ð f ð x Þ ¼ a 0 x þ a 1 x 2 2 þ a 2 x 3 3 þ … The Power Series that results from the term-by-term integration will converge to the integral of the original function (within the interval of convergence of the original series), although this is not necessarily true at the endpoints of the interval. This can be a powerful method for handling dif fi cult integrals. Even if the integration of the function f ( x ) is dif fi cult, the integration of the individual terms of its powers series expansion will always be simple, as this only involves integrating integer powers of x . If we need to fi nd the derivative of f ( x ), we can simply differentiate the Power Series of f ( x ) term by term to produce f 0 ð x Þ ¼ a 1 þ 2 a 2 x þ 3 a 2 x 2 þ … The resulting series will converge to the derivative of the original function within the original interval of convergence. Once again, this is not necessarily true at the endpoints of the interval. And if we want to evaluate the function f ( x ) 56 Power Series

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  Boyce's Elementary Differential Equations and Boundary Value Problems

  • William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  In the case of division, the radius of convergence of the resulting Power Series may be less than . 190 CHAPTER 5 Series Solutions of Second-Order Linear Equations 8. The function f is continuous and has derivatives of all orders for |x − x 0 | < . Moreover, f ′ , f ′′ , . . . can be computed by differentiating the series termwise; that is, f ′ (x) = a 1 + 2a 2 (x − x 0 ) + ⋅ ⋅ ⋅ + na n (x − x 0 ) n−1 + ⋅ ⋅ ⋅ = ∞ ∑ n=1 na n (x − x 0 ) n−1 , f ′′ (x) = 2a 2 + 6a 3 (x − x 0 ) + ⋅ ⋅ ⋅ + n(n − 1)a n (x − x 0 ) n−2 + ⋅ ⋅ ⋅ = ∞ ∑ n=2 n(n − 1)a n (x − x 0 ) n−2 , and so forth, and each of the series converges absolutely for |x − x 0 | < . 9. The value of a n is given by a n = f (n) (x 0 ) n! . The series is called the Taylor 1 series for the function f about x = x 0 . 10. If ∞ ∑ n=0 a n (x − x 0 ) n = ∞ ∑ n=0 b n (x − x 0 ) n for each x in some open interval with center x 0 , then a n = b n for n = 0, 1, 2, 3, . . . . In particular, if ∞ ∑ n=0 a n (x − x 0 ) n = 0 for each such x, then a 0 = a 1 = ⋅ ⋅ ⋅ = a n = ⋅ ⋅ ⋅ = 0. A function f that has a Taylor series expansion about x = x 0 f (x) = ∞ ∑ n=0 f (n) (x 0 ) n! (x − x 0 ) n , with a radius of convergence  > 0, is said to be analytic at x = x 0 . All of the familiar functions of calculus are analytic except perhaps at certain easily recognized points. For example, sin x and e x are analytic everywhere, 1∕x is analytic except at x = 0, and tan x is analytic except at odd multiples of ∕2. According to statements 6 and 7, if f and g are analytic at x 0 , then f ± g, f ⋅ g, and f ∕g (provided that g(x 0 ) ≠ 0) are also analytic at x = x 0 . In many respects the natural context for the use of Power Series is the complex plane. The methods and results of this chapter nearly always can be directly extended to differential equations in which the independent and dependent variables are complex-valued.

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  Elementary Differential Equations and Boundary Value Problems

  • William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  In most cases the coefficients   can be most easily obtained by equating coefficients in the equivalent relation ∞ ∑ =0   ( −  0 )  = ( ∞ ∑ =0   ( −  0 )  )( ∞ ∑ =0   ( −  0 )  ) = ∞ ∑ =0 (  ∑ =0    − )( −  0 )  . In the case of division, the radius of convergence of the resulting Power Series may be less than . 5.1 Review of Power Series 197 8. The function  is continuous and has derivatives of all orders for | −  0 | < . Moreover,  ′ ,  ′′ , … can be computed by differentiating the series termwise; that is,  ′ () =  1 + 2 2 ( −  0 ) + ⋯ +   ( −  0 ) −1 + ⋯ = ∞ ∑ =1   ( −  0 ) −1 ,  ′′ () = 2 2 + 6 3 ( −  0 ) + ⋯ + ( − 1)  ( −  0 ) −2 + ⋯ = ∞ ∑ =2 ( − 1)  ( −  0 ) −2 , and so forth, and each of the series converges absolutely for | −  0 | < . 9. The value of   is given by   =  () ( 0 ) ! . The series is called the Taylor 1 series for the function  about  =  0 . 10. If ∞ ∑ =0   ( −  0 )  = ∞ ∑ =0   ( −  0 )  for each  in some open interval with center  0 , then   =   for  = 0, 1, 2, 3, … . In particular, if ∞ ∑ =0   ( −  0 )  = 0 for each such , then  0 =  1 = ⋯ =   = ⋯ = 0. A function  that has a Taylor series expansion about  =  0 () = ∞ ∑ =0  () ( 0 ) ! ( −  0 )  , with a radius of convergence  > 0, is said to be analytic at  =  0 . All of the familiar functions of calculus are analytic except perhaps at certain easily recognized points. For example, sin() and   are analytic everywhere, 1 ∕  is analytic except at  = 0, and tan() is analytic except at odd multiples of  ∕ 2. According to statements 6 and 7, if  and  are analytic at  0 , then  ± ,  ⋅ , and  ∕  (provided that ( 0 ) ≠ 0) are also analytic at  =  0 . In many respects the natural context for the use of Power Series is the complex plane.

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  Functions of a Complex Variable

  • Hemant Kumar Pathak, Ravi Agarwal, Yeol Je Cho(Authors)
  • 2015(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 3 Power Series and Elementary Functions 3.1 Power Series ...................................................... 123 3.1.1 Absolute Convergence of Power Series .................. 124 3.1.2 Some Special Tests for Convergence of Series ........... 124 3.2 Certain Theorems on Power Series .............................. 125 3.2.1 Abel’s Theorem .......................................... 125 3.2.2 Cauchy–Hadamard’s Theorem .......................... 126 3.2.3 Circle and Radius of Convergence of a Power Series .... 127 3.2.4 Analyticity of Sum Functions of Power Series .......... 129 3.2.5 Abel’s Limit Theorem ................................... 131 3.3 Elementary Functions of a Complex Variable ................... 141 3.4 Many-Valued Functions: Branches ............................... 151 3.5 Logarithms and Power Functions ................................ 153 3.6 The Riemann Surface for Log z .................................. 160 3.7 Historical Remarks ............................................... 171 3.1 Power Series We discuss here certain fundamental results concerning infinite series in pow-ers of the complex variable z called Power Series . In subsequent sections, we shall introduce elementary functions. To understand the nature of elementary functions, we require a clear understanding of these results. A Power Series 1 is an infinite series of the type ∞ X n =0 a n z n or ∞ X n =0 a n ( z -z 0 ) n , (3 . 1) where variable z and the constants a 0 , z 0 are, in general, complex numbers and a n in independent of z. As a special case, z , a n and z 0 may be real. By a change of variable z = ξ + z 0 , the second form of Power Series reduces to 1 The term “Power Series” alone usually refers to a series of the form (1), but does not include series of negative terms of z or z -z 0 such as a 1 z -1 + a 2 z -2 + · · · . 123 124 Functions of a Complex Variable ∞ ∑ n =0 a n ξ n .

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  Infinite Sequences and Series

  • Konrad Knopp(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  Chapter 4

  Power Series

  4.1.   The circle of convergence

  We have already encountered several times, series of the form Σ

  aν z

  ν , where z has been permitted to be arbitrary to a certain extent. Such series, and, somewhat more generally, series of the form Σa ν (z–z 0 )ν where z 0 is a fixed number, are called Power Series . In what follows, there is usually no loss of generality in considering only Power Series of the first form. For if we set z–z 0 = z ′ for abbreviation, and then drop the accent, the second form goes over into the first.

  Examples of such series were

  The first converges if, and only if, |z |<1, i .e ., in the interior of the unit circle. The third converges for every z , i.e ., “in the entire plane”. Finally, is an example of a Power Series that converges only for z = 0, because, for z ≠ 0, νν z ν = (νz )ν obviously does not tend to 0.

  We shall show, first of all, that every Power Series possesses an analogous convergence behavior, i .e ., that it converges either in the entire plane, or in a certain circle about 0 as center, or only for z = 0. Indeed, we have

  Theorem 1. Let Σ

  aν z

  ν be an arbitrary Power Series, and set .

  Then ,

  a) for a = 0, the series is everywhere convergent ,

  b) for a = + ∞, the series is divergent for every z≠ 0.

  c) If finally , 0<α< + ∞,

  then the series is absolutely convergent for every z with , divergent for every z with |z| > r

  . (The behavior of the series on the circumference |z | = r can then be quite varied; see below.) Thus we have in all three cases, with suitable interpretation ,

  PROOFS . a) α = 0 means that , because . Hence, if z is an arbitrary number, then also . The assertion now follows from the radical test.

  b) Let a = + ∞ and z ≠ 0, so that |z | > 0. Then, according to 2.2 ,5 , or infinitely often, and consequently Σ

  aν z

  ν is divergent.

  c) In this case, let z be an arbitrary, but henceforth fixed, number, with . Choose a positive number ρ for which , and hence

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  The Calculus Lifesaver

  All the Tools You Need to Excel at Calculus

  • Adrian Banner(Author)
  • 2009(Publication Date)
  • Princeton University Press

    (Publisher)

  C h a p te r 26 Taylor and Power Series: How to Solve Problems In this chapter, we’ll look at how to solve four different classes of problems involving Taylor series, Taylor polynomials and Power Series: • how to find where Power Series converge or diverge; • how to manipulate Taylor series to get other Taylor series or Taylor polynomials; • using Taylor series or Taylor polynomials to find derivatives; and • using Maclaurin series to find limits. 26.1 Convergence of Power Series Let’s say we have a Power Series about x = a : ∞ X n =0 a n ( x -a ) n . As we saw in the case of geometric series, a Power Series might converge for some x and diverge for other x . The question that we want to ask is this: given our Power Series, for which x does it converge, and for which x does it diverge? Furthermore, if the series converges for a specific x , it would be nice to know whether the convergence is absolute or merely conditional. So, let’s see what could possibly happen, and then we’ll take advantage of these observations. 26.1.1 Radius of convergence We want to find out for which x the Power Series ∑ ∞ n =0 a n ( x -a ) n converges. On the face of it, it seems like we have to answer infinitely many questions here, since there are infinitely many values of x to substitute in and test to see whether the series converges or not. Let’s draw a number line representing different values of x . For each x such that our Power Series converges, we’ll put a check mark above it, whereas if the Power Series diverges for a particular x , we’ll put a cross instead. (Of course, we won’t do this for every single x , 552 • Taylor and Power Series: How to Solve Problems since the diagram would get crowded! We’ll just do enough to get the idea.) For example, the geometric series ∑ ∞ n =0 x n converges when -1 < x < 1 and diverges otherwise, so its picture looks like this: -1 0 1 Note that I took special care to indicate the divergence at the endpoints 1 and -1.

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  Elementary Theory of Analytic Functions of One or Several Complex Variables

  • Henri Cartan(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  CHAPTER I

  Power Series in One Variable

  1.     Formal Power Series

  1.     ALGEBRA OF POLYNOMIALS

  Let K be a commutative field. We consider the formal polynomials in one symbol (or ‘indeterminate’) X with coefficients in K (for the moment we do not give a value to X). The laws of addition of two polynomials and of multiplication of a polynomial by a ‘scalar’ makes the set K[X] of polynomials into a vector space over K with the infinite base

  Each polynomial is a finite linear combination of the X

  n

  with coefficients in K and we write it , where it is understood that only a finite number of the coefficients a

  n

  are non-zero in the infinite sequence of these coefficients. The multiplication table

  defines a multiplication in K[X]; the product

  is , where

  This multiplication is commutative and associative. It is bilinear in the sense that

  for all polynomials P, P1 P2 , Q, and all scalars λ. It admits as unit element (denoted by 1) the polynomial such that a0 = 1 and an = 0 for n > 0. We express all these properties by saying that K[X], provided with its vector space structure and its multiplication, is a commutative algebra with a unit element over the field K; it is, in particular, a commutative ring with a unit element.

  2.     THE ALGEBRA OF FORMAL SERIES

  A formal Power Series in X is a formal expression , where this time we no longer require that only a finite number of the coefficients a

  n

  are non-zero. We define the sum of two formal series by

  and the product of a formal series with a scalar by The set K[[X]] of formal series then forms a vector space over K. The neutral element of the addition is denoted by 0; it is the formal series with all its coefficients zero.

  The product of two formal series is defined by the formula (1.1 ), which still has a meaning because the sum on the right hand side is over a finite number of terms. The multiplication is still commutative, associative and bilinear with respect to the vector structure. Thus K[[X]] is an algebra over the field K with a unit element (denoted by 1), which is the series such that a0 = 1 and an = 0 for n

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  The Fundamentals of Mathematical Analysis

  • G. M. Fikhtengol'ts, I. N. Sneddon, M. Stark, S. Ulam(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  For instance, starting from the known expansion . x^ Λ:* X^ [Sec. 253, (11)], we find ? ^ 1 x^ I x^ 1 Jc2« + i e-x^dx = X 1 ... + ( -1) h .... J 1! 3 ^ 2! 5 ^ ^, 2n + l ^ 0 Similar expansions can successfully be used for approximate calculations and for forming tables of values for integrals not expressible in finite form. 276. Termwise dijfferentíatíon of a Power Series (6) The Power Series (1) within its interval of convergence can be differentiated termwise, so that for the sum f{x) of the series there exists a deriva-tive and can be expressed thus: 00 fx) = ^na,x^-^ = a^ + 2a^x+ ... + WÖ.X-^ + .... ( 8 ) n = l Whatever value oí χ = X q, — R < X q< R, we take, we can choose two numbers r^ and r such that | I < < '* < ^ · In ^i^w of the convergence of the series (5) [Sec. 273], its general term is bounded: | a j . r « < L (« = 1 , 2 , 3 , ... ;L = const). Then, when | Λ:| [n= 1 , 2 , 3 , . . . ; ^0 = 7-= constj § 3. Power Series 95 r I n -l + the convergence of which is easy to ascertain with the help of d'Alembert's test (taking into account that r^jr < 1).

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  An Introduction to Mathematical Analysis

  International Series of Monographs on Pure and Applied Mathematics

  • Robert A. Rankin, I. N. Sneddon, S. Ulam, M. Stark(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER 5 FUNCTIONS DEFINED BY Power Series 21. GENERAL THEORY OF Power Series 21.1. Radius of convergence. A series of the form Sa n z n ( N> 0), ( 1 ) n, N where z and the coefficients a„ are real or complex numbers, is called a Power Series. In most of the general results that we prove we shall take N = 0; there is no loss of generality in doing this, since we can put any Power Series with positive initial suffix N into this form by defining S „ to be zero for 0 <— n < N. When the notation a„zn is used in this chapter it is to be understood that the initial suffix N is not negative. It is clear that every Power Series is convergent when z = 0. It can happen that a Power Series converges for no other values of z. Thus the series S( nz)n considered in 18.2F does not converge if z == 0; n,1 this follows by the same argument as given there for real z = C. Other Power Series converge only for points z in some bounded region of the complex plane (see § 16.1). Thus we saw in 17.1F that the geometric series S zn is convergent only for points z in the interior n, O of the unit circle, i.e. for z I < 1. Finally, some Power Series converge everywhere, i.e. for all complex Z. This is the case for the series S( z/n)n considered in 18.2 F, since the n, 1 argument given there, with x = j z ~, shows that the series is abso-lutely convergent for all Z. When we say that the Power Series (1) converges at a point z a we mean that S a„zó converges. Similarly, if .9 2 is a set of real or complex numbers, the statement that the series (1) converges on U means that SS „zn converges for all z E ~. n, N 191 192 AN INTRODUCTION TO MATHEMATICAL ANALYSIS THEOREM 21.1.1 If S an o converges, where z o + 0, then S anzn is absolutely convergent for all z with z j < j Z 0 ~ . Proof. Since Z an ó converges, a n zo and therefore 1anz0 ~ tends to zero as n -± cc. Hence there exists an integer N such that a n zój < 1 for all n >_ N.

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  Approximate Analytical Methods for Solving Ordinary Differential Equations

  • T.S.L Radhika, T. Iyengar, T. Rani(Authors)
  • 2014(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Section 2.4 gives a brief remark on the series solutions at irregular singular points (IRSPs). The Taylor series method is discussed in Section 2.5. Also provided are a list of articles and books for ready reference together with a list 8 APPROXIMATE ANALYTICAL METHODS of research articles that provide greater insight on the applications of the method discussed. Definition An infinite series of the form a x x n n n ∑ -= ∞ ( ) 0 0 (2.1) where a n ’s are constants, is called a Power Series about the point (cen-tered) x x = 0 . In particular, if x = 0 0 , Equation (2.1) is said to be a Power Series centered at the origin. Definition : Interval of Convergence and Radius of Convergence If there exists a positive real number R such that the series Equation (2.1) is convergent for x x R -< 0 and divergent for x x R -> 0 , then R is called the radius of convergence of the Power Series, and the interval R R ( ) -, is called the interval of the conver-gence of the series Equation (2.1). In fact, it will be the interval of absolute convergence of the Power Series. Note: The radius of convergence R can be calculated by the formula R a a n n n = →∞ + li m . 1 This radius of convergence R can be zero, finite, or infinite as well. For rigorous proof, refer to some standard works on real analysis (Apostol, 1974; Rudin, 1976). Definition If f x ( ) has a Power Series expansion of the form a x x n n n ( ) ∑ -= ∞ 0 0 with radius of convergence R , then f x ( ) is said to be analytic at x x = 0 with R as the radius of analyticity. 9 Power Series METHOD Let us look at some examples for which we calculate the radius of convergence of some Power Series. Example 2.1 For the series n x R n n n n n n n n n n ∑ ( ) ( ) --= -+ -= = ∞ →∞ + + + 3 5 3 , lim ( 3) .( 1).5 ( 3) . .5 5 3 n+1 0 2 1 1 Hence, the series has a radius of convergence R = 5 3 , and the interval of convergence of this Power Series is ( ) -5/3,5/3 .

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