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We remark that we have used the equality|zn | = |z|n here.Example 4 The series 1 + z + z2 + · · · converges absolutely for |z| < 1 and its sum1is s = 1−z. For |z| ≥ 1 it does not converge, since in that case the general term doesnot tend to zero.Series of the formc0 + c1 (z − z0 ) + · · · + cn (z − z0 )n + · · ·are called power series.By applying the Cauchy criterion (Sect.
3.1.4) to the series|c0 | + c1 (z − z0 ) + · · · + cn (z − z0 )n + · · · ,(5.113)(5.114)5.5 Complex Numbers and Elementary Functions269we conclude that this series converges if# −1|z − z0 | < lim n |cn | ,n→∞and that the general term does not tend to zero if |z − z0 | ≥ (limn→∞From this we obtain the following proposition.√n|cn |)−1 .Proposition 3 (The Cauchy–Hadamard23 formula) The power series (5.113) converges inside the disk |z − z0 | < R with center at z0 and radius given by the Cauchy–Hadamard formulaR=1limn→∞√n|cn |.(5.115)At any point exterior to this disk the power series diverges.At any point interior to the disk, the power series converges absolutely.Remark In regard to convergence on the boundary circle |z − z0 | = R Proposition 3is silent, since all the logically admissible possibilities really can occur.Examples The series∞ nz ,5)n=1∞ 1 n6)n=1 n z ,and∞7)n=11 nzn2converge in the unit disk |z| < 1, but the series 5) diverges at every point z where|z| = 1.
The series 6) diverges for z = 1 and (as one can show) converges for z = −1.The series 7) converges absolutely for |z| = 1, since | n12 zn | = n12 .One must keep in mind the possible degenerate case when R = 0 in (5.115),which was not taken account of in Proposition 3. In this case, of course, the entiredisk of convergence degenerates to the single point z0 of convergence of the series(5.113).The following result is an obvious corollary of Proposition 3.Corollary (Abel’s first theorem on power series) If the power series (5.113) converges at some value z∗ , then it converges, and indeed even absolutely, for any valueof z satisfying the inequality |z − z0 | < |z∗ − z0 |.The propositions obtained up to this point can be regarded as simple extensionsof facts already known to us.
We shall now prove two general propositions about23 J.Hadamard (1865–1963) – well-known French mathematician.2705Differential Calculusseries that we have not proved up to now in any form, although we have partlydiscussed some of the questions they address.Proposition 4 If a series z1 + z2 + · · · + zn + · · · of complex numbers convergesabsolutely, then a series zn1 + zn2 + · · · + znk + · · · obtained by rearranging24 itsterms also converges absolutely and has the same sum.∞Proof Using the convergence∞ of the series n=1 |zn |, given a number ε > 0, wechoose N ∈ N such that n=N +1 |zn | < ε.We then find an index K ∈ N such that all the terms in the sum SN = z1 +· · ·+zNare among the terms of the sum s̃k = zn1 + · · · + znk for k > K. If s = ∞n=1 zn , wefind that for k > K|s − s̃k | ≤ |s − sN | + |sN − s̃k | ≤∞|zn | +n=N +1∞|zn | < 2ε.n=N +1Thus we have shown that s̃k → s as k → ∞.
If we apply what has just beenproved to the series |z1 | + |z2 | + · · · + |zn | + · · · and |zn1 | + |zn2 | + · · · + |znk | +· · · , we find that the latter series converges. Thus Proposition 4 is now completelyproved.Our next proposition will involve the product of two series(a1 + a2 + · · · + an + · · · ) · (b1 + b2 + · · · + bn + · · · ).The problem is that if we remove the parentheses and form all possible pair-wiseproducts ai bj , there is no natural order for summing these products, since we havetwo indices of summation. The set of pairs (i, j ), where i, j ∈ N, is countable, aswe know. Therefore we could write down a series having the products ai bj as termsin some order. The sum of such a series might depend on the order in which theseterms are taken.
But, as we have just seen, in absolutely convergent series the sumis independent of any rearrangement of the terms. Thus, it is desirable to determinewhen the series with terms ai bj converges absolutely.Proposition 5 The product of absolutely convergent series is an absolutely convergent series whose sum equals the product of the sums of the factor series.Proof We begin by remarking that whatever finite sum ai bj of terms of the formai bj we take, we can always find N such that the product of the sums AN = a1 +· · · + aN and BN = b1 + · · · + bN contains all the terms in that sum.
ThereforeNNN∞∞ |ai bj | ≤ai bj ≤|ai bj | =|ai | ·|bj | ≤|ai | · ||bj |,i,j =124 Thei=1j =1i=1j =1term with index k in this series is the term znk with index nk in the original series. Here themapping N k → nk ∈ N is assumed to be a bijective mapping on the set N.5.5 Complex Numbers and Elementary Functions271from which it follows that the series ∞i,j =1 ai bj converges absolutely and that itssum is uniquely determined independently of the order of the factors. In that case thesum can be obtained, for example, as the limit of the products of the sums An = a1 +· · · + an and Bn = b1 + · · · + bn . But An Bn → AB as n → ∞, where A = ∞n=1 anand B = ∞b,whichcompletestheproofofProposition5.n=1 nThe following example is very important.∞ 1 m∞ 1 nconverge absolutely. In theExample 8 The seriesn=0 n! a andm=0 m! bproduct of these series let us group together all monomials of the form a n bm havingthe same total degree n + m = k.
We then obtain the series∞ k=0 n+m=k1 n 1 mab .n! m!Butm+n=k1 n mk!11a b =a n bk−n = (a + b)k ,n!m!k!n!(k − n)!k!kn=0and therefore we find that∞∞∞1 n 1 m 1a ·b =(a + b)k .n!m!k!n=0m=0(5.116)k=05.5.3 Euler’s Formula and the Connections Amongthe Elementary FunctionsIn Examples 1)–3) we established the absolute convergence in C of the series obtained by extending into the complex domain the Taylor series of the functions ex ,sin x, and cos x, which are defined on R. For that reason, the following definitionsare natural ones to make for the functions ez , cos z, and sin z in C:111z + z2 + z3 + · · · ,1!2!3!11cos z := 1 − z2 + z4 − · · · ,2!4!11sin z := z − z3 + z5 − · · · .3!5!ez = exp z := 1 +(5.117)(5.118)(5.119)2725Differential CalculusFollowing Euler,25 let us make the substitution z = iy in Eq. (5.117).
By suitablygrouping the terms of the partial sums of the resulting series, we find that11111(iy) + (iy)2 + (iy)3 + (iy)4 + (iy)5 + · · · =1!2!3!4!5!1 2 1 411 3 1 5= 1 − y + y − ··· + iy − y + y − ··· ,2!4!1!3!5!1+that is,eiy = cos y + i sin y.(5.120)This is the famous Euler formula.In deriving it we used the fact that i 2 = −1, i 3 = −i, i 4 = 1, i 5 = i, and soforth.
The number y in formula (5.120) may be either a real number or an arbitrarycomplex number.It follows from the definitions (5.118) and (5.119) thatcos(−z) = cos z,sin(−z) = − sin z,that is, cos z is an even function and sin z is an odd function. Thuse−iy = cos y − i sin y.Comparing this last equality with formula (5.120), we obtain1 iye + e−iy ,21 iye − e−iy .sin y =2icos y =Since y is any complex number, it would be better to rewrite these equalitiesusing notation that leaves no doubt of this fact:1 ize + e−iz ,21 ize − e−iz .sin z =2icos z =(5.121)Thus, if we assume that exp z is defined by relation (5.117), then formulas(5.121), which are equivalent to the expansions (5.118) and (5.119), like the for25 L.
Euler (1707–1783) – eminent mathematician and specialist in theoretical mechanics, of Swissextraction, who lived the majority of his life in St. Petersburg. In the words of Laplace, “Euler isthe common teacher of all mathematicians of the second half of the eighteenth century.”5.5 Complex Numbers and Elementary Functions273mulas1 ze + e−z ,21 zsinh z = e − e−z ,2cosh y =(5.122)can be taken as the definitions of the corresponding circular and hyperbolic functions. Disregarding all the considerations about trigonometric functions that led usto this step, which have not been rigorously justified (even though they did leadus to Euler’s formula), we can now perform a typical mathematical trick and takeformulas (5.121) and (5.122) as definitions and obtain from them in a completelyformal manner all the properties of the circular and trigonometric functions.For example, the fundamental identitiescos2 z + sin2 z = 1,cosh2 z − sinh2 z = 1,like the parity properties, can be verified immediately.The deeper properties, such as, for example, the formula for the cosine and sineof a sum follow from the characteristic property of the exponential function:exp(z1 + z2 ) = exp(z1 ) · exp(z2 ),(5.123)which obviously follows from the definition (5.117) and formula (5.116).
Let usderive the formulas for the cosine and sine of a sum:On the one hand, by Euler’s formulaei(z1 +z2 ) = cos(z1 + z2 ) + i sin(z1 + z2 ).(5.124)On the other hand, by the property of the exponential function and Euler’s formulaei(z1 +z2 ) = eiz1 eiz2 = (cos z1 + i sin z1 )(cos z2 + i sin z2 ) == (cos z1 cos z2 − sin z1 sin z2 ) + i(sin z1 cos z2 + cos z2 sin z2 ). (5.125)If z1 and z2 were real numbers, then, equating the real and imaginary parts of thenumbers in formulas (5.124) and (5.125), we would now have obtained the requiredformulas. Since we are trying to prove them for any z1 , z2 ∈ C, we use the fact thatcos z is even and sin z is odd to obtain yet another equality:e−i(z1 +z2 ) = (cos z1 cos z2 − sin z1 sin z2 ) − i(sin z1 cos z2 + cos z1 sin z2 ). (5.126)Comparing (5.125) and (5.126), we findcos(z1 + z2 ) =1 i(z1 +z2 )e+ e−i(z1 +z2 ) = cos z1 cos z2 − sin z1 sin z2 ,22745sin(z1 + z2 ) =Differential Calculus1 i(z1 +z2 )e− e−i(z1 +z2 ) = sin z1 cos z2 + cos z1 sin z2 .2iThe corresponding formulas for the hyperbolic functions cosh z and sinh z couldbe obtained in a completely analogous manner.
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