1) The function f (z) := x3y2+ix2y3 is, according to 2, real-differentiable throughout C. The Cauchy-Riemann equations hold at the point c = (a, b) exactly when 3a2b2 = 3a2b2 and 2a3b = -2ab3, i.e., when ab(a2 + b2) = 0;
which, since a, b are real, amounts to ab = 0. In summary, the points at which f is complex-differentiable are the points on the two coordinate axes.
2) We will assume the reader is acquainted with the real exponential function e° and the meal trigonometric functions, cost, sin t, t E R. The
function
e(z) := ex cosy + iex sin y
is real-differentiable at every z = x+iy in C, by 2, and the Cauchy-Riemann equations clearly hold at every point. Thus e(z) is complex-differentiable in C and e'(z) = ux(z)+ivx(z) = e(z). In 5. 1.1 we will see that e(z) is the complex exponential function exp z = Eo V, .
3) This example is for readers acquainted with the real logarithm function log t, t > 0, and the real arctangent function arctan t, t E R; the notation refers to the principal branch, that is, the values of arctangent lying between -7r/2 and 7r/2. From 2 and the properties of these functions, namely the identities log'(t) = t-1 and arctan'(t) = (1 + t2)-1, we see that
l(z) :=2 log(x2 + y2) + i arctanb
is real-differentiable throughout C \ {z E C : Rz = 0} and satisfies the Cauchy-Riemann equations there as well. Thus £(z) is complex-differenti- able everywhere to the left and to the right of the imaginary axis. A direct calculation shows that
?(z)=ux(z)+ivx(z)= zEC with tzO0.
z
In 5.4.4 we will see that (z) coincides in the right half-plane with the principal branch of the complex logarithm function and that
2(z) = log z = E( (z-1)' for z E B1(1).
1
v
In the last two examples complex-differentiable functions were fashioned out of transcendental real functions with the help of the Cauchy-Riemann equations. However, in this book - as in classical function theory generally - this mode of constructing complex-differentiable functions will not be pursued any further.
4) If f = u + iv is complex-differentiable in D, then throughout D
1f'12 = det ` vx vy u2 + v2 = uy + v2,
x y
a fact which follows from 1 f'12 = f177 = us + vs on account of u., = vy, uy = -vx. I f'(z)12 is thus the value of the Jacobian functional determinant of the mapping (x, y) H (u(x, y), v(x, y)); this determinant is never negative and in fact is positive at every point z E D where f'(z) # 0. In example 2) we see, e.g., that
1e (z)12 = e2x cost y + e2x sin2 y = e23R:
4*. Harmonic functions. Not all real-valued real-differentiable functions u(x, y) occur as real parts of complex-differentiable functions. The Cauchy- Riemann equations lead at once to a quite restrictive necessary condition on u for this to happen. To formulate it, recall that the twice continuous (real-) differentiability of u in D means that the partial derivatives ux and uy are differentiable and the four second-order partial derivatives uxx, uxy, uyx, uyy are continuous in D. As a consequence of this continuity uxy = uyx in D, another well-known fact from the real differential calculus, often proved via the Mean Value Theorem. Now the aforementioned necessary condition reads
Theorem. If f = u + iv is complex-differentiable in D and if u and v are twice continuously real-differentiable in D, then
uxx + uyy = 0 ,vxx + vyy = 0 in D.
Proof. Because f is complex-differentiable throughout D, ux = vy and uy = -vx in D. More partial differentiation yields uxx = vyx, uxy = vyy,
uyy = -vy, uyx = -vxx.
It follows that uxx + uyy = vyx - vxy and vxx + vyy = -uyx + uxy in D. Since all the second-order partial derivatives of u and v are continuous in D, we have, as noted above, that uxy = uyx and vxy = vyx; and so the claimed equalities follow.The supplemental assumption of the twice continuous differentiability of u and v in this theorem is actually superfluous, because it turns out that ev- ery complex-differentiable function is infinitely often complex-differentiable (cf. 7.4.1).
In the literature the differential polynomial
D:=a
y822is known as the Laplace operator. For every twice real-differentiable func- tion u : D -' R the function Du = uxx + uyy, is defined in D. u is called a potential function in D if u satisfies the potential-equation Du = 0 in D.
(The language is motivated by considerations from physics, especially elec- trostatics, because functions with Du = 0 arise as potentials in physics.) Potential-functions are also known as harmonic functions.
The essence of the theorem is that the real and imaginary parts of complex-differentiable functions are potential-functions. Thus simple ex- amples of potential-functions can be obtained from the examples in the preceding section; e.g., !`z2 = 2xy, $2z3 = x3 - 3xy2 are harmonic in C.
Furthermore, the functions
2e(z)
= ex cosy ,
£3 (z) = ex sin y R (z) = log I z I ,&(z) = arctan
are harmonic in their domains of definition. The function x2 + y2 = Iz12 is not harmonic and so not the real part of any complex-differentiable function. (Take a look at x2 - y2 = $tz2; alternatively, look at 0(x2 + y2).) 0 For every harmonic polynomial u(x, y) E R[x, y] one can directly write down a complex polynomial p(z) E C[z] whose real part is u(x, y); namely, p(z) := 2u(Z z, 2i z) -u(0, 0). The reader should clarify this for himself with a few examples; he might even give a proof of the general assertion.
Harmonic functions of two variables played a big role in classical mathematics and gave essential impulses to it. In this connection let us only recall here the famous
DIRIcHLET Boundary-Value Problem. A real-valued continuous function g on the boundary 8E = {z E C : Izj = 1} of the unit disc is given. A continuous function u on E U 8E is sought having the properties
that uIOE = g and uIE is a potential-function in E.
It can be shown that there is always exactly one such function u.
The theory of holomorphic (see next section for the definition) functions has gotten valuable stimulus from the theory of harmonic functions. Some properties
of harmonic functions (integral formulas, maximum principle, convergence theo- rems, etc.) are shared by holomorphic functions. But nowadays it is customary to develop the theory of holomorphic functions completely and then to derive from it the fundamental properties of harmonic functions of two variables.
Exercises
Exercise 1. Where are the following functions complex-differentiable?
a) f (x + iv) = x4y5 + ixy3 b) f (x + iy) = y2 sin x + iy
c) f (x + iy) = sin2(x + y) + i cos2(x + y)
d) f (x + iy) = -6(cos x + i sin x) + (2 - 2i)y3 + 15(y2 + 2y).
Exercise 2. Let G be a region in C and f = u+iv be complex-differentiable in G. Show that a function v : G -r R satisfies u+iv complex-differentiable
in G, if and only if, v -,b is constant.
Exercise 3. For each of the given functions u : C - R find all functions v : C -+ R such that u + iv is complex-differentiable:
a) u(x + iy) = 2x3 - 6xy2 + x2 - y2 - y b) u(x + iy) = x2 - y2 + e-y sin x - ey cos x.
Exercise 4. Show that for integer n > 1 the function u : Cx R, z '-+ log Iz"I is harmonic but is not the real part of any function which is complex-differentiable in C x .
Exercise 5. Show that every harmonic function u : C R is the real part of some complex-differentiable function on C.
§3 Holomorphic functions
Now we introduce the fundamental idea of all of function theory. A func- tion f : D -+ C is called holomorphic in the domain D if f is complex- differentiable at every point of D; we say f is holomorphic at c E D if there is an open neighborhood U of c lying in D such that the restriction f JU of f to U is holomorphic in U.
The set of all points at which a function is holomorphic, is always open in C. A function which is holomorphic at c is complex-differentiable at c
but a function which is complex-differentiable at c need not be holomorphic at c. For example, the function
1(z) := x3 y2 + ix2y3 , where z = x + iy ; x, y E R,
is, according to 2.3, complex-differentiable at the points of the coordinate axes but nowhere else. So this function is not holomorphic at any point of
C.
The set of all holomorphic functions in the domain D is always denoted by O(D). We naturally have the inclusions
C c O(D) C C(D);
the first because constant functions are (complex-) differentiable every- where in C and the second because complex-differentiability implies conti- nuity.
1. Differentiation rules are proved as in the case of real-differentiation;
doing so provides some evidence that the definition of complex-differenti- ability in use today offers considerable advantages over Riemann's defini- tion via his differential equations.
Sum- and Product-rule. Let f : D --+ C and g : D -+ C be holomorphic in D. Then for all a, b E C the functions a f + bg and f g are holomorphic
in D, with
(af + bg)' = af' + bg' (sum-rule),
(f g)' = f'9 + fg' (product-rule).
We will be content to recall how the proof of the product-rule goes. By hypothesis, for each c E D there are functions fc, 9, : D --+ C which are continuous at c and satisfy
f (z) = f (c) + (z - c) fc(z) , g(z) = g(c) + (z - c)gc(z) , z E D.
Multiplication yields, for all z E D,
(f . 9)(z) = (f 9)(c) + (z - c)Ih(z)9(c) + f (c)9g(z) + (z - c)(.ff . 9c)(z)].
Since the square-bracketed expression is a function of z E D which is evidently continuous at c, the complex-differentiability at c of the product function f g is confirmed, with moreover (f g)'(c) being the value of that function at c, viz.,
(f '9)'(c) = fc(c)9(c) + f(c)9c(c) = .f'(c)9(c) + f(c)9 (c)
From the sum- and product-rules follows, as with real-differentiability:
Every complex polynomial p(z) = ao + al z + + a,,z' E C[z] is holo- morphic in C and satisfies p(z) = al + 2a2z + - - + E C[z].
As in the reals, we also have a
Quotient rule. Let f, g be holomorphic and g zero-free in D. Then the quotient function
9
: D -+ C is holomorphic in D and
(:)'
= f ' 9 1 9 ' (Quotient-rule).9 92
Differentiation of the composite function h o g is codified in the Chain-rule. Let g E O(D), h E O(D') be holomorphic functions with g(D) C D'. Then the composite function h o g : D - C is holomorphic in D and
(h o g)'(z) = h'(g(z)) g'(z), z E D (Chain-rule).
The quotient- and chain-rules are proved just as for real-differentiability.
On the basis of Theorem 2.1 a function f = u + iv is holomorphic in the domain D C C exactly when f is real-differentiable and satisfies the Cauchy- Riemann equations u= = vy, uy = -v= throughout D. But these differentiability hypotheses may be dramatically weakened. For example, we have
A continuous function f : D C is already holomorphic in D if through each point c E D there are two distinct straight lines L, L' along which the limits
lim f (z) - f (c) lim f (z) - f (c)
zEL, z-.c z- C zEL', z-.c z- C
exist and are equal.
This theorem is due to D. MENCHOFF: "Sur Is generalisation des conditions de Cauchy-Riemann," Fund. Math. 25(1935), 59-97. As a special case, that in which every L is parallel to the x-axis and every L' is parallel to the y-axis, we have the so-called LOOMAN-MENCHOFF theorem:
A continuous function f : D C is already holomorphic in D if the partial derivatives u1, uy, v=, vu of the real-valued functions u := tf, v := £ f exist and satisfy the Cauchy-Riemann equations u. = vy, uy = -v= throughout D.
The hypothesis about the continuity of f, or some weaker surrogate, is needed, as the following example shows:
f (z) := exp(-z-4) for z E C" , f (0) := 0.
(At the trouble-point z = 0, the two partial derivatives of f exist and are 0 by an elementary use of the Mean Value Theorem of real analysis, since
ae (x) _ 4x-5f(x) (x 0 0] and By(iy) = 4y-3f (y) (y # 0) imply that lim.-o '(x) _ limy-0 2L(iy) = 0.) On the other hand, if we ask that f be continuous through- out D but only ask that the Cauchy-Riemann equations hold at one point, then complex-differentiability at that point cannot be inferred, as the example
f (Z) := IzI-4z5 for z E C" , f(0) := 0 shows.
Actually the result which is usually designated as the Looman-Menchoff the- orem contains even weaker differentiability hypotheses than those stated above:
the partial derivatives need only exist on a set whose complement in D is count- able and the Cauchy-Riemann equations need only hold on a set whose com- plement in D has area 0. A very accessible proof of this, together with a full history and bibliography of other possible weakenings of the differentiability hy- potheses will be found in J. D. GRAY and S. A. MoRRis, "When is a function that satisfies the Cauchy-Riemann equations analytic?", Amer. Math. Monthly 85(1978), 246-256. Another elementary account, which deals with Menchoff's first theorem as well, is K. MEIER, "Zum Satz von Looman-Menchoff," Comm.
Math. Hely. 25(1951), 181-195; some simplifications of this paper will be found in M. G. ARSOVE, "On the definition of an analytic function," Amer. Math.
Monthly 62(1955), 22-25.
2. The C-algebra O(D). The differentiation rules yield directly that:
For every domain D in C the set O(D) of all functions which are holo- morphic in D is a C-subalgebra of the C-algebra C(D). The units of O(D) are exactly the zero free functions.
For the exponential function e(z) and the logarithm function £Sz) of examples 2) and 3), respectively, in 2.3 we have e(z) E O(C) and £(z) E O(C \ iR).
In contrast to C(D), the C-algebra O(D) does not contain the conjugate f of each of its functions f; we saw, e.g., that z E O(D) but z §E O(D).
Also in general, if f E O(D) then none of elf, 8°f of If I belongs to O(D);
for example, each of tz, and IzI is not complex-differentiable at any point of C.
Every polynomial in z is holomorphic in C; every rational function (meaning quotient of polynomials) is holomorphic in the complement of the zero-set of its denominator. Further examples of holomorphic functions can only be secured via limiting processes and so are no longer considered ele- mentary functions. In 4.3.2 we will see that power series inside their circles of convergence furnish an inexhaustible reservoir of holomorphic functions.
If f is a holomorphic function in D, then
f':D-C,z-f'(z)
is another function defined on D. It is called the (first) derivative of f in D. If one thinks about differentiable functions on R like xIxI, then there is no reason to expect f' to be holomorphic in D. But a fundamental theorem of function theory, which we will get from the Cauchy integral formula (but not until 7.4.1), says exactly this, that f' is holomorphic in D whenever f is. As a consequence, every holomorphic function in D turns out to be infinitely often (complex-) differentiable in D, that is, all the derivatives f', . .. , f (r" ), exist. Here, as in the reals, we understand by the mth derivative f (m) of f (in case it exists) the first derivative of f(m-1), m = 1, 2, ...; thus f (0) := f and f(m)
(f
The same proof (induction) used for functions on R will also establish Leibniz' product-rule for higher derivatives of holomorphic functions(f . g)(m) _
V!
f
(k)g(t).k+t=m
3.
Characterization of locally constant functions.
The following statements about a function f : D -+ C are equivalent:i) f is locally constant in D.
ii) f is holomorphic in D and f'(z) = 0 for all z E D.
First Proof. Only ii) = i) needs to be verified. Let u := Rf, v := 3'f.
Since f' = ux + ivx and ux = vy, V. = -uy, our hypothesis ii) means that ux(z) = uy(z) = vx(z) = vy(z) = 0 for all z E D. From a well-known theorem of real analysis it then follows that each of u and v, and therewith also f = u + iv, is locally constant in D.
The theorem from real analysis used above is proved via the Mean Value Theo- rem, but its use can easily be circumvented by another elementary (compactness) argument:
Second proof. Consider any B = Br(b) C D and any z E B. Let L denote the line segment from b to z and let e > 0 be given. For each c E L there is a disc B6(c) C D, b = b(c) > 0, such that (cf. 1.1 and remember that f' a 0):
If(w)-f(c)I <- elw - cl for all w E Bs(c).
Because finitely many of the discs B6(c) suffice to cover the compactum L, there is a succession of points zo = z on L such that
It follows that
If (z.,) - f(z.,-1)I <- flz - zp- II,
1<v<n.
n n
if (z) - f(b)I = I if (Z.) - if(zO - f(z.-=)I
1 1
n
<
elz-z-1I=elz-bl.
Here e > 0 is arbitrary, so f (z) = f (b) follows. This is true for each z E B, that is, fIB is constant.
After studying complex integral calculus (cf. 6.3.2), we will give a third proof of this theorem, using primitives. On the basis of 0.6 the theorem can also be expressed thus:
For a region G in C, a function f : G -+ C is constant in G if and only if it is holomorphic in G and f' vanishes everywhere in G.
We will illustrate the result of this paragraph by two examples.
1) Every function f which is holomorphic in D and assumes only real values, respectively, only purely imaginary values, is locally constant in D.
Proof. In case u := R f = f, we have v := 3 f = 0 and so the Cauchy- Riemann equations give ux = vy = 0 = vx and so f' = ux + ivx = 0 in D.
By the theorem f is locally constant in D. If, on the other hand, we have f = i3` f throughout D, then we apply what we just learned to if in the role of f to conclude the local constancy of f.
2) Every holomorphic function which has constant modulus in D is locally constant in D.
Proof. Suppose f = u+iv is holomorphic in D and u2+v2 = c is constant in D. Then differentiation of this equation with respect to y gives uuy+vvy = 0 and so, since uy = -vx, uvx = vvy. Since also uux +vvx = 0 and ux = vy, we get
0 =
(u2 +v 2 )u" = cux.
Similarly, cvx = 0. If c = 0 then, of course, f is constant (equal 0) in D.
If c 0 0, we now have f' = ux + ivx = 0 in D, so that f is locally constant in D by the theorem.
In 8.5.1 we will prove the Open Mapping Theorem for holomorphic func- tions; this theorem contains both of the above examples as trivial cases.
4. Historical remarks on notation. The word "holomorphic" was in- troduced in 1875 by BRIOT and BOUQUET, [BB], 2nd ed., p.14. In their
1st edition (cf. pp. 3,7 and 11) they used instead of "holomorphic" the designation "synectic", which goes back to CAUCHY. Other synonyms in the older literature are "monogenic", "monodromic", "analytic" and "reg- ular". These and other terms originally described various properties, like having vanishing integral over every closed curve (cf. Chapter 6) or having local power series expansions or satisfying the Cauchy-Riemann equations, etc., and so were not at first recognized as synonyms. When the theory of functions reached maturity, these properties were all seen to be equivalent (and the reader will see this presently); so it is appropriate that most of these terms have now faded into oblivion. "Analytic" is still sometimes used as a synonym for "holomorphic", but usually it has a more technical meaning having to do with the Weierstrass continuation process.
Actually, as late as 1851 CAUCHY still had no exact definition of the class of functions for which his theory was valid "La theorie des fonctions de variables imaginaires presente des questions delicates qu'il importait de resoudre . (The theory of functions of an imaginary variable presents delicate questions which it was important to resolve .)"; thus begins a Comptes Rendus note on February 10, 1851 bearing the title "Sur les fonctions de variables imaginaires" ((Fuvres (1) 11, pp. 301-304). See pp.
169, 170 of BOTTAZZINI [H4).
The notation O(D) is used - since about 1952 - by the French school around Henri CARTAN, especially in the function theory of several variables. It is some- times said that 0 was chosen to honor the great Japanese mathematician OKA, and it is sometimes even maintained that the 0 reflects the French pronuncia- tion of the word holomorphic. Nevertheless, the choice of the symbol 0 appears to have been purely accidental. In a letter of March 22, 1982 to the author of this book, H. CARTAN wrote: "Je m'etais simplement inspires d'une notation utilisese par van der Waerden dans son classique trait6 `Modern Algebra' (cf.
par exemple §16 de la 2` edition allemande, p. 52)". [1 was simply inspired by a notation used by van der Waerden in his classic treatise `Modern Algebra' (cf.
for example, §16 Vol. I of the English translation).]
Exercises
Exercise 1. Let G be a region in C. Determine all holomorphic functions f on G for which (f)2 + i(!0f )2 is also holomorphic on G.
Exercise 2. Suppose that f = u + iv is holomorphic in the region G C C and that for some pair of non-zero complex numbers a and b, au + by is constant in G. Show that then f itself is constant in G.
Exercise 3. Let f = it + iv be holomorphic in the region G and satisfy it = h o v for some differentiable function h : R -* R. Show that f is constant.
Exercise 4. Let D, D' be domains in C, g : D --+ C continuous with g(D) c D', and h E O(D'). Show that if h' is zero-free on g(D) and h o g is holomorphic in D, then g is holomorphic in D. Hint: For fixed c E D consider the C-linearization of h o g at c and that of h at g(c).
§4 Partial differentiation with respect to x, y,
zand z
If f : D -+ C is real-differentiable at c and T = T f (c), then the limit relation
lim If (c + h) - f (c) - T(h) I = 0
h-+o IhI
is valid without the absolute value signs. (But in general, for functions into R', stripping away the absolute values results in a meaningless division by the vector h.) Upon setting z = c + h, this observation becomes
Differentiability criterion. f : D --+ C is real-differentiable at c precisely when there exist a (uniquely determined) R-linear map T : C -+ C and a function f : D - C, which is continuous at c with i(c) = 0, such that
f(z) = f(c)+T(h)+hf(z).
If we write the R-linear differential
T(h)= (v= (c) vy(c)) of f = u + iv at c in the form
(1) T(h) =T(1)Rh+T(i)!alh
or in the form
(2) T(h) = Ah+µh,
then we are led almost automatically to introduce, besides the partial derivatives of u and v with respect to x and y, also the partial deriva- tives of f itself with respect to x and y and even with respect to z and z (subsection 1 below). Because, thanks to 0.1.2, there are among the quantities ux(c), , v,,(c),T(1),T(i), A, p the relations