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Incidentally, as can be seen fromformulas (5.121) and (5.122), these functions are connected with cos z and sin z bythe relationscosh z = cos iz,sinh z = −i sin iz.However, to obtain even such geometrically obvious facts as the equalitysin π = 0 or cos(z + 2π) = cos z from the definitions (5.121) and (5.122) is very difficult. Hence, while striving for precision, one must not forget the problems wherethese functions naturally arise. For that reason, we shall not attempt at this pointto overcome the potential difficulties connected with the definitions (5.121) and(5.122) when describing the properties of the trigonometric functions.

We shall return to these functions after presenting the theory of integration. Our purpose atpresent was only to demonstrate the remarkable unity of seemingly completely different functions, which would have been impossible to detect without going into thedomain of complex numbers.If we take as known that for x ∈ Rcos(x + 2π) = cos x,sin(x + 2π) = sin x,cos 0 = 1,sin 0 = 0,then from Euler’s formula (5.120) we obtain the relationeiπ + 1 = 0,(5.127)in which all the most important constants of the different areas of mathematics arerepresented: 1 (arithmetic), π (geometry), e (analysis), and i (algebra).From (5.123) and (5.127), as well as from (5.120), one can see thatexp(z + i2π) = exp z,that is, the exponential function is a periodic function on C with the purely imaginary period T = i2π .Taking account of Euler’s formula, we can now represent the trigonometric notation (5.105) for a complex number in the formz = reiϕ ,where r is the modulus of z and ϕ its argument.The formula of de Moivre now becomes very simple:zn = r n einϕ .(5.128)5.5 Complex Numbers and Elementary Functions275Fig.

5.25Fig. 5.26Fig. 5.275.5.4 Power Series Representation of a Function. AnalyticityA function w = f (z) of a complex variable z with complex values w, defined ona set E ⊂ C, is a mapping f : E → C. The graph of such a function is a subsetof C × C = R2 × R2 = R4 , and therefore is not visualizable in the traditional way.To compensate for this loss to some extent, one usually keeps two copies of thecomplex plane C, indicating points of the domain of definition in one and points ofthe range of values in the other.In the examples below the domain E and its image under the correspondingmapping are indicated.Example 9 See Fig. 5.25.Example 10 See Fig.

5.26.Example 11 See Fig. 5.27.These correspondences follow from the equalities i = eiπ/2 , z = reiϕ , and iz =that is, a rotation through angle π2 has occurred.rei(r+π/2) ,Example 12 See Fig. 5.28.2765Differential CalculusFig. 5.28Fig. 5.29Fig. 5.30Fig. 5.31For, if z = reiϕ , then z2 = r 2 ei2ϕ .Example 13 See Fig. 5.29.Example 14 See Fig.

5.30.It is clear from Examples 12 and 13 that under this function the unit disk mapsinto itself, but is covered twice.Example 15 See Fig. 5.31.If z = reiϕ , then by (5.128), we have zn = r n einϕ , so that in this case the imageof the disk of radius r is the disk of radius r n , each point of which is the image of npoints in the original disk (located, as it happens, at the vertices of a regular n-gon).5.5 Complex Numbers and Elementary Functions277The only exception is the point w = 0, whose pre-image is the point z = 0. However, as z → 0, the function zn is an infinitesimal of order n, and so we say that atz = 0 the function has a zero of order n.

Taking account of this kind of multiplicity,one can now say that the number of pre-images of every point w under the mapping z → zn = w is n. In particular, the equation zn = 0 has the n coincident rootsz1 = · · · = zn = 0.In accordance with the general definition of continuity, a function f (z) ofa complex variable is called continuous at a point z0 ∈ C if for any neighborhood V (f (z0 )) of its value f (z0 ) there exists a neighborhood U (z0 ) such thatf (z) ∈ V (f (z0 )) for all z ∈ U (z0 ). In short,lim f (z) = f (z0 ).z→z0The derivative of a function f (z) at a point z0 , as for the real-valued case, isdefined asf (z0 ) = limz→z0f (z) − f (z0 ),z − z0(5.129)if this limit exists.The equality (5.129) is equivalent tof (z) − f (z0 ) = f (z0 )(z − z0 ) + o(z − z0 )(5.130)as z → z0 , corresponding to the definition of differentiability of a function at thepoint z0 .Since the definition of differentiability in the complex-valued case is the sameas the corresponding definition for real-valued functions and the arithmetic properties of the fields C and R are the same, one may say that all the general rules fordifferentiation hold also in the complex-valued case.Example 16(f + g) (z) = f (z) + g (z),(f · g) (z) = f (z)g(z) + f (z)g (z),(g ◦ f ) (z) = g f (z) · f (z),so that if f (z) = z2 , then f (z) = 1 · z + z · 1 = 2z, or if f (z) = zn , then f (z) =nzn−1 , and ifPn (z) = c0 + c1 (z − z0 ) + · · · + cn (z − z0 )n ,thenPn (z) = c1 + 2c2 (z − z0 ) + · · · + ncn (z − z0 )n−1 .2785Differential CalculusnTheorem 1 The sum f (z) = ∞n=0 cn (z − z0 ) of a power series is an infinitelydifferentiable function inside the entire disk in which convergence occurs.

Moreover,f (k) (z) =∞dk cn (z − z0 )n ,kdzk = 0, 1, . . . ,n=0andcn =1 (n)f (z0 ),n!n = 0, 1, . . . .Proof The expressions for the coefficient follows in an obvious way from the expressions for f (k) (z) for k = n and z = z0 .As for the formula for f (k) (z), it suffices to verify this formula for k = 1, sincethe function f (z) will then be the sum of a power series.n−1 is indeed theThus, let us verify that the function ϕ(z) = ∞n=1 ncn (z − z0 )derivative of f (z).We begin by remarking that by the Cauchy–Hadamard formula (5.115) the radiusof convergence of the derived series is the same as the radius of convergence R ofthe original power series for f (z).For simplicityof notation fromwe shall assume that z0 = 0, that∞ now onnn−1 and that these series converge foris, f (z) = ∞n=0 cn z , ϕ(z) =n=1 ncn z|z| < R.Since a power series converges absolutely on the interior of its disk of convern−1n−1 ≤gence, we note (and this is crucial) that the estimate∞ |ncn zn−1 | = n|cn ||z|n−1holds for |z| ≤ r < R, and that series n=1 n|cn |rconverges.

Hence,n|cn |rfor any ε > 0 there exists an index N such that ∞∞ εn−1 ncn z ≤ncn r n−1 ≤3n=N +1n=N +1for |z| ≤ r.Thus at any point of the disk |z| < r the function ϕ(z) is within 3ε of the N thpartial sum of the series that defines it.Now let ζ and z be arbitrary points of this disk. The transformation∞f (ζ ) − f (z) ζ n − zn==cnζ −zζ −zn=1=∞cn ζ n−1 + ζ n−2 z + · · · + ζ zn−2 + zn−1n=1and the estimate |cn (ζ n−1 + · · · + zn−1 )| ≤ |cn |nr n−1 enable us to conclude, asabove, that the difference quotient we are interested in is equal within 3ε to the5.5 Complex Numbers and Elementary Functions279partial sum of the series that defines it, provided |ζ | < r and |z| < r. Hence, for|ζ | < r and |z| < r we have NN ζ n − z n f (ζ ) − f (z)εn−1 ≤−ϕ(z)−cncz+2 .nnζ −zζ −z3n=1n=1If we now fix z and let ζ tend to z, passing to the limit in the finite sum, we seethat the right-hand side of this last inequality will be less than ε for ζ sufficientlyclose to z, and hence the left-hand side will be also.Thus, for any point z in the disk |z| < r < R, we have verified that f (z) = ϕ(z).Since r is arbitrary, this relation holds for any point of the disk |z| < R.This theorem enables us to specify the class of functions whose Taylor seriesconverge to them.A function is analytic at a point z0 ∈ C if it can be represented in a neighborhoodof the point in the following (“analytic”) form:f (z) =∞cn (z − z0 )n ,n=0that is, as the sum of a power series in z − z0 .It is not difficult to verify (see Problem 7 below) that the sum of a power seriesis analytic at any interior point of the disk of convergence of the series.Taking account of the definition of analyticity, we deduce the following corollaryfrom the definition of analyticity.Corollary a) If a function is analytic at a point, then it is infinitely differentiable atthat point, and its Taylor series converges to it in a neighborhood of the point.b) The Taylor series of a function defined in a neighborhood of a point and infinitely differentiable at that point converges to the function in some neighborhoodof the point if and only if the function is analytic.In the theory of functions of a complex variable one can prove a remarkable factthat has no analogue in the theory of functions of a real variable.

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