proof The of the converse -- that any function whose real and imaginary components satisfy the CauchyRiemann equations is differentiable -- will be omitted, but can be found in any basic text on complex analysis, e.g., [3, 65, 118]. Remark : It is worth pointing out that equation (7.19) tells us that f satisfies f /x = - i f /y, which, reassuringly, agrees with the first equation in (7.5). Example 7.4. Consider the elementary function z 3 = (x3 - 3 x y 2 ) + i (3 x2 y - y 3 ). Its real part u = x3 - 3 x y 2 and imaginary part v = 3 x2 y - y 3 satisfy the CauchyRiemann equations (7.18), since v u = 3 x2 - 3 y 2 = , x y u v = -6xy = - . y x
Theorem 7.3 implies that f (z) = z 3 is complex differentiable. Not surprisingly, its derivative turns out to be f (z) = u v v u +i = -i = (3 x2 - 3 y 2 ) + i (6 x y) = 3 z 2 . x x y y
Fortunately, the complex derivative obeys all of the usual rules that you learned in real-variable calculus. For example, d n z = n z n-1 , dz d cz e = c ec z , dz 1 d log z = , dz z (7.20)
and so on. The power n can be non-integral -- or even, in view of the identity z n = en log z , complex, while c is any complex constant. The exponential formulae (7.14) for the complex trigonometric functions implies that they also satisfy the standard rules d cos z = - sin z, dz d sin z = cos z. dz (7.21)
The formulae for differentiating sums, products, ratios, inverses, and compositions of complex functions are all identical to their real counterparts, with similar proofs. Thus, thankfully, you don't need to learn any new rules for performing complex differentiation! Remark : There are many examples of seemingly reasonable functions which do not have a complex derivative. The simplest is the complex conjugate function f (z) = z = x - i y. Its real and imaginary parts do not satisfy the CauchyRiemann equations, and hence z does not have a complex derivative. More generally, any function f (z, z) that explicitly depends on the complex conjugate variable z is not complex-differentiable. 1/19/12 225
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Power Series and Analyticity A remarkable feature of complex differentiation is that the existence of one complex derivative automatically implies the existence of infinitely many! All complex functions f (z) are infinitely differentiable and, in fact, analytic where defined. The reason for this surprising and profound fact will, however, not become evident until we learn the basics of complex integration in Section 7.6. In this section, we shall take analyticity as a given, and investigate some of its principal consequences. Definition 7.5. A complex function f (z) is called analytic at a point z0 C if it has a power series expansion
f (z) = a0 + a1 (z - z0 ) + a2 (z - z0 ) + a3 (z - z0 ) + = which converges for all z sufficiently close to z0 .
2
3
n=0
an (z - z0 )n ,
(7.22)
Typically, the standard ratio or root tests for convergence of (real) series that you learned in ordinary calculus, [7, 129], can be applied to determine where a given (complex) power series converges. We note that if f (z) and g(z) are analytic at a point z0 , so is their sum f (z) + g(z), product f (z) g(z) and, provided g(z0 ) = 0, ratio f (z)/g(z). Example 7.6. All of the real power series found in elementary calculus carry over to the complex versions of the functions. For example,
e = 1+z + z + z + =
z
1 2
2
1 6
3
n=0
zn n!
(7.23)
is the power series for the exponential function based at z0 = 0. A straightforward application of the ratio test proves that the series converges for all z. On the other hand, the power series 1 = 1 - z2 + z4 - z6 + = (-1)k z 2 k , (7.24) 2+1 z
k=0
converges inside the unit disk, where | z | < 1, and diverges where outside, | z | > 1. Again, convergence is established through the ratio test. The ratio test is inconclusive when | z | = 1, and we shall leave the more delicate question of precisely where on the unit disk this complex series converges to a more advanced treatment, e.g., [3]. In general, there are three possible options for the domain of convergence of a complex power series (7.22): (a) The series converges for all z. (b) The series converges inside a disk | z - z0 | < of radius > 0 centered at z0 and diverges for all | z - z0 | > outside the disk. The series may converge at some (but not all) of the points on the boundary of the disk where | z - z0 | = . (c) The series only converges, trivially, at z = z0 . The number is known as the radius of convergence of the series. In case (a), we say = , while in case (c), = 0, and the series does not represent an analytic function. 1/19/12 226
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An example determining converges is delegated to
that has = 0 is the power series n! z n . In the intermediate case (b), precisely where on the boundary of the convergence disk the power series quite delicate, and will not be pursued here. The proof of this result is Exercise ; see also [3, 65, 118] for further details.
Remarkably, the radius of convergence for the power series of a known analytic function f (z) can be determined by inspection, without recourse to any fancy convergence tests! Namely, is equal to the distance from z0 to the nearest singularity of f (z), meaning a point where the function fails to be analytic. In particular, the radius of convergence = if and only if f (z) is analytic for all z, with no singularities; examples include ez , cos z, and sin z. On the other hand, the rational function f (z) = z2 1 1 = +1 (z + i )(z - i )
has singularities at z = i , and so its power series (7.24) has radius of convergence = 1, which is the distance from z0 = 0 to the singularities. Thus, the extension of the theory of power series to the complex plane serves to explain the apparent mystery of why, as a real function, (1 + x2 )-1 is well-defined and analytic for all real x, but its power series only converges on the interval ( -1, 1 ). It is the complex singularities that prevent its convergence when | x | > 1. If we expand (z 2 + 1)-1 in a power series at some other point, say z0 = 1 + 2 i , then we need to determine which singularity is closest. We compute | i - z0 | = | -1 - i | = 2, while | - i - z0 | = | -1 - 3 i | = 10, and so the radius of convergence = 2 is the smaller. Thus we can determine the radius of convergence without any explicit formula for its (rather complicated) Taylor expansion at z0 = 1 + 2 i . There are, in fact, only three possible types of singularities of a complex function f (z): Pole. A singular point z = z0 is called a pole of order n > 0 if and only if f (z) = h(z) , (z - z0 )n (7.25)
Essential singularity. By definition, a singularity is essential if it is not a pole or a branch point. Essential singularities are considerably more complicated than poles and branch points. The quintessential example is the essential singularity of the function e1/z at z0 = 0. Example 7.7. The complex function f (z) = 1/19/12 ez ez = z3 - z2 - 5 z - 3 (z - 3)(z + 1)2 227
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Branch point. We have already encountered the two basic types: algebraic branch points, such as the function n z at z0 = 0, and logarithmic branch points such as log z at z0 = 0. The degree of the branch point is n in the first case and in the second.
where h(z) is analytic at z = z0 and h(z0 ) = 0. The simplest example of such a function is f (z) = a (z - z0 )-n for a = 0 a complex constant.