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Chapter 6
Functions of a
Complex Variable I
Analytic Properties Mapping
The imaginary numbers are a wonderful flight of God’s spirit; they are almost an amphibian between being and not being.
—Gottfried Wilhelm von Leibniz, 1702
The theory of functions of one complex variable contains some of the most powerful and widely useful tools in all of mathematical analysis. To indicate why complex variables are important, we mention briefly several areas of application. First, for many pairs of functions u and v, both u and v satisfy Laplace’s equation in two real dimensions
∇^2 u =
∂^2 u ( x , y ) ∂ x^2
∂^2 u ( x , y ) ∂ y^2
For example, either u or v may be used to describe a two-dimensional electrostatic potential. The other function then gives a family of curves or- thogonal to the equipotential curves of the first function and may be used to describe the electric field E. A similar situation holds for the hydrodynamics of an ideal fluid in irrotational motion. The function u might describe the velocity potential, whereas the function v would then be the stream function. In some cases in which the functions u and v are unknown, mapping or transforming complex variables permits us to create a (curved) coordinate system tailored to the particular problem. Second, complex numbers are constructed (in Section 6.1) from pairs of real numbers so that the real number field is embedded naturally in the com- plex number field. In mathematical terms, the complex number field is an extension of the real number field, and the latter is complete in the sense that 318
6.1 Complex Algebra 319
any polynomial of order n has n (in general) complex zeros. This fact was first proved by Gauss and is called the fundamental theorem of algebra (see Sections 6.4 and 7.2). As a consequence, real functions, infinite real series, and integrals usually can be generalized naturally to complex numbers simply by replacing a real variable x , for example, by complex z. In Chapter 8, we shall see that the second-order differential equations of interest in physics may be solved by power series. The same power series may be used by replacing x by the complex variable z. The dependence of the solution f ( z ) at a given z 0 on the behavior of f ( z ) elsewhere gives us greater insight into the behavior of our solution and a powerful tool (analytic continuation) for extending the region in which the solution is valid. Third, the change of a parameter k from real to imaginary transforms the Helmholtz equation into the diffusion equation. The same change trans- forms the Helmholtz equation solutions (Bessel and spherical Bessel func- tions) into the diffusion equation solutions (modified Bessel and modified spherical Bessel functions). Fourth, integrals in the complex plane have a wide variety of useful applications:
- (^) Evaluating definite integrals (in Section 7.2)
- (^) Inverting power series
- (^) Infinite product representations of analytic functions (in Section 7.2)
- (^) Obtaining solutions of differential equations for large values of some vari- able (asymptotic solutions in Section 7.3)
- (^) Investigating the stability of potentially oscillatory systems
- (^) Inverting integral transforms (in Chapter 15)
Finally, many physical quantities that were originally real become complex as a simple physical theory is made more general. The real index of refraction of light becomes a complex quantity when absorption is included. The real energy associated with an energy level becomes complex, E = m ± i , when the finite lifetime of the level is considered. Electric circuits with resistance R , capacitance C , and inductance L typically lead to a complex impedance Z = R + i (ω L − (^) ω^1 C ). We start with complex arithmetic in Section 6.1 and then introduce complex functions and their derivatives in Section 6.2. This leads to the fundamental Cauchy integral formula in Sections 6.3 and 6.4; analytic continuation, singu- larities, and Taylor and Laurent expansions of functions in Section 6.5; and conformal mapping, branch point singularities, and multivalent functions in Sections 6.6 and 6.7.
6.1 Complex Algebra
As we know from practice with solving real quadratic equations for their real zeros, they often fail to yield a solution. A case in point is the following example.
6.1 Complex Algebra 321
A graphical representation is a powerful means to see a complex number or variable. By plotting x (the real part of z ) as the abscissa and y (the imaginary part of z ) as the ordinate, we have the complex plane or Argand plane shown in Fig. 6.1. If we assign specific values to x and y , then z corresponds to a point ( x , y ) in the plane. In terms of the ordering mentioned previously, it is obvious that the point ( x , y ) does not coincide with the point ( y , x ) except for the special case of x = y. Complex numbers are points in the plane, and now we want to add, sub- tract, multiply, and divide them, just like real numbers. All our complex variable analyses can now be developed in terms of ordered pairs 1 of numbers ( a , b ), variables ( x , y ), and functions ( u ( x , y ), v( x , y )). We now define addition of complex numbers in terms of their Cartesian components as z 1 + z 2 = ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 + x 2 , y 1 + y 2 ) = z 2 + z 1 , (6.2) that is, two-dimensional vector addition. In Chapter 1, the points in the xy - plane are identified with the two-dimensional displacement vector r = xˆ x + yˆ y. As a result, two-dimensional vector analogs can be developed for much of our complex analysis. Exercise 6.1.2 is one simple example; Cauchy’s theorem (Section 6.3) is another. Also, − z + z = (− x , − y ) + ( x , y ) = 0 so that the negative of a complex number is uniquely specified. Subtraction of complex numbers then proceeds as addition: z 1 − z 2 = ( x 1 − x 2 , y 1 − y 2 ). Multiplication of complex numbers is defined as z 1 z 2 = ( x 1 , y 1 ) · ( x 2 , y 2 ) = ( x 1 x 2 − y 1 y 2 , x 1 y 2 + x 2 y 1 ). (6.3) Using Eq. (6.3) we verify that i^2 = (0, 1) · (0, 1) = (−1, 0) = −1 so that we can also identify i =
−1 as usual, and further rewrite Eq. (6.1) as z = ( x , y ) = ( x , 0) + (0, y ) = x + (0, 1) · ( y , 0) = x + iy. (6.4) Clearly, the i is not necessary here but it is truly convenient and tra- ditional. It serves to keep pairs in order—somewhat like the unit vectors of vector analysis in Chapter 1. With complex numbers at our disposal, we can determine the complex zeros of z^2 + z + 1 = 0 in Example 6.1.1 as z = − 12 ± 2 i
3 so that
z^2 + z + 1 =
z +
i 2
z +
i 2
factorizes completely.
Complex Conjugation
The operation of replacing i by – i is called “taking the complex conjugate.” The complex conjugate of z is denoted by z ∗, 2 where z ∗^ = x − iy. (6.5)
(^1) This is precisely how a computer does complex arithmetic. (^2) The complex conjugate is often denoted by ¯ z in the mathematical literature.
322 Chapter 6 Functions of a Complex Variable I
y (^) z
q q
x
( x , y )
z* ( x , − y )
Figure 6. Complex Conjugate Points
The complex variable z and its complex conjugate z ∗^ are mirror images of each other reflected in the x -axis; that is, inversion of the y -axis (compare Fig. 6.2). The product zz ∗^ leads to
zz ∗^ = ( x + iy )( x − iy ) = x^2 + y^2 = r^2. (6.6)
Hence,
( zz ∗)^1 /^2 = | z |
is defined as the magnitude or modulus of z. Division of complex numbers is most easily accomplished by replacing the denominator by a positive number as follows:
z 1 z 2
x 1 + iy 1 x 2 + iy 2
( x 1 + iy 1 )( x 2 − iy 2 ) ( x 2 + iy 2 )( x 2 − iy 2 )
( x 1 x 2 + y 1 y 2 , x 2 y 1 − x 1 y 2 ) x 22 + y 22
which displays its real and imaginary parts as ratios of real numbers with the same positive denominator. Here, | z 2 |^2 = x 22 + y^22 is the absolute value (squared) of z 2 , and z ∗ 2 = x 2 − iy 2 is called the complex conjugate of z 2. We write | z 2 |^2 = z ∗ 2 z 2 , which is the squared length of the associated Cartesian vector in the complex plane. Furthermore, from Fig. 6.1 we may write in plane polar coordinates
x = r cos θ, y = r sin θ (6.8)
and
z = r (cos θ + i sin θ). (6.9)
In this representation r is the modulus or magnitude of
z ( r = | z | = ( x^2 + y^2 )^1 /^2 )
324 Chapter 6 Functions of a Complex Variable I
The choice of polar representation [Eq. (6.10)] or Cartesian representation, [Eqs. (6.1) and (6.4)] is a matter of convenience. Addition and subtraction of complex variables are easier in the Cartesian representation [Eq. (6.2)]. Multiplication, division, powers, and roots are easier to handle in polar form [Eqs. (6.8)–(6.10)]. Let us examine the geometric meaning of multiplying a function by a com- plex constant.
EXAMPLE 6.1.3 Multiplication by a Complex Number^ When we multiply the complex variable z by i = e i π/^2 , for example, it is rotated counterclockwise by 90◦^ to iz = ix − y = (− y , x ). When we multiply z = re i θ^ by e i α^ , we get re i (θ+α)^ , which is z rotated by the angle α.
Similarly, curves defined by f ( z ) = const. are rotated when we multiply a function by a complex constant. When we set
f ( z ) = ( x + iy )^2 = ( x^2 − y^2 ) + 2 ixy = c = c 1 + ic 2 = const.,
we define two hyperbolas
x^2 − y^2 = c 1 , 2 xy = c 2.
Upon multiplying f ( z ) = c by a complex number Ae i α^ , we obtain
Ae i α^ f ( z ) = A (cos α + i sin α)( x^2 − y^2 + 2 ixy ) = A [ i (2 xy cos α + ( x^2 − y^2 ) sin α) − 2 xy sin α + ( x^2 − y^2 ) cos α].
The hyperbolas are scaled by the modulus A and rotated by the angle α. ■
Analytically or graphically, using the vector analogy, we may show that the modulus of the sum of two complex numbers is no greater than the sum of the moduli and no less than the difference (Exercise 6.1.3):
| z 1 | − | z 2 | ≤ | z 1 + z 2 | ≤ | z 1 | + | z 2 |. (6.12)
Because of the vector analogy, these are called the triangle inequalities. Using the polar form [Eq. (6.8)] we find that the magnitude of a product is the product of the magnitudes,
| z 1 · z 2 | = | z 1 | · | z 2 |. (6.13)
Also,
arg( z 1 · z 2 ) = arg z 1 + arg z 2. (6.14)
From our complex variable z complex functions f ( z ) or w( z ) may be con- structed. These complex functions may then be resolved into real and imagi- nary parts
w( z ) = u ( x , y ) + i v( x , y ), (6.15)
6.1 Complex Algebra 325
y v
z -plane
x
1
2
w -plane
u
1
2
Figure 6.
The Function w ( z ) = u ( x , y ) + iv ( x , y ) Maps Points in the xy -Plane into Points in the uv -Plane
in which the separate functions u ( x , y ) and v( x , y ) are pure real. For example, if f ( z ) = z^2 , we have
f ( z ) = ( x + iy )^2 = x^2 − y^2 + 2 ixy.
The real part of a function f ( z ) will be labeled f ( z ), whereas the imag- inary part will be labeled I f ( z ). In Eq. (6.15),
w( z ) = u ( x , y ), Iw( z ) = v( x , y ). (6.16)
The relationship between the independent variable z and the dependent vari- able w is perhaps best pictured as a mapping operation. A given z = x + iy means a given point in the z -plane. The complex value of w( z ) is then a point in the w-plane. Points in the z -plane map into points in the w-plane, and curves in the z -plane map into curves in the w-plane, as indicated in Fig. 6.3.
Functions of a Complex Variable
All the elementary (real) functions of a real variable may be extended into the complex plane, replacing the real variable x by the complex variable z. This is an example of the analytic continuation mentioned in Section 6.5. The extremely important relations, Eqs. (6.4), (6.8), and (6.9), are illustrations. Moving into the complex plane opens up new opportunities for analysis.
EXAMPLE 6.1.4 De Moivre’s Formula^ If Eq. (6.11) is raised to the^ n th power, we have
e in θ^ = (cos θ + i sin θ) n. (6.17)
Using Eq. (6.11) now with argument n θ, we obtain
cos n θ + i sin n θ = (cos θ + i sin θ) n. (6.18)
This is De Moivre’s formula.
6.1 Complex Algebra 327
L
R C
V 0 sin wt
Figure 6. Electric RLC Circuit with Alternating Current
EXAMPLE 6.1.6 Electric Circuits^ An electric circuit with a current^ I^ flowing through a resistor and driven by a voltage V is governed by Ohm’s law, V = I R , where R is the resistance. If the resistance is replaced by an inductance L , then the voltage and current are related by V = L dIdt. If the inductance is replaced by the capacitance C , then the voltage depends on the charge Q of the capacitor: V = QC. Taking the time derivative yields C dVdt = dQdt = I. Therefore, a circuit with a resistor, a coil, and a capacitor in series (see Fig. 6.4) obeys the ordinary differential equation
L
dI dt
Q
C
- I R = V = V 0 cos ω t (6.21)
if it is driven by an alternating voltage with frequency ω. In electrical engineer- ing it is a convention and tradition to use the complex voltage V = V 0 e i ω t^ and a current I = I 0 e i ω t^ of the same form, which is the steady-state solution of Eq. (6.21). This complex form will make the phase difference between current and voltage manifest. At the end, the physically observed values are taken to be the real parts (i.e., V 0 cos ω t = V 0 e i ω t^ , etc). If we substitute the exponential time dependence, use dI / dt = i ω I , and integrate I once to get Q = I / i ω in Eq. (6.21), we find the following complex form of Ohm’s law :
i ω LI +
I
i ω C
+ RI = V ≡ Z I.
We define Z = R + i (ω L − (^) ω^1 C ) as the impedance, a complex number, obtaining V = Z I , as shown.
More complicated electric circuits can now be constructed using the impe- dance alone—that is, without solving Eq. (6.21) anymore—according to the following combination rules:
- (^) The resistance R of two resistors in series is R = R 1 + R 2.
- (^) The inductance L of two inductors in series is L = L 1 + L 2.
- (^) The resistance R of two parallel resistors obeys (^1) R = (^) R^1 1 +^
1 R 2.
- (^) The inductance L of two parallel inductors obeys (^) L^1 = (^) L^1 1 +^
1 L 2.
- (^) The capacitance of two capacitors in series obeys (^) C^1 = (^) C^1 1 +^
1 C 2.
- (^) The capacitance of two parallel capacitors obeys C = C 1 + C 2.
328 Chapter 6 Functions of a Complex Variable I
In complex form these rules can be stated in a more compact form as follows:
- (^) Two impedances in series combine as Z = Z 1 + Z 2 ;
- (^) Two parallel impedances combine as (^1) Z = (^) Z^1 1 +^
1 Z 2.^ ■
Complex numbers extend the real number axis to the complex number plane so that any polynomial can be completely factored. Complex numbers add and subtract as two-dimensional vectors in Cartesian coordinates:
z 1 + z 2 = ( x 1 + iy 1 ) + ( x 2 + iy 2 ) = x 1 + x 2 + i ( y 1 + y 2 ).
They are best multiplied or divided in polar coordinates of the complex plane:
( x 1 + iy 1 )( x 2 + iy 2 ) = r 1 e i θ^1 r 2 e i θ^2 = r 1 r 2 e i (θ^1 +θ^2 )^ , r^2 j = x^2 j + y^2 j , tan θ (^) j = yj / x (^) j.
The complex exponential function is given by e z^ = e x (cos y + i sin y ). For z = x + i 0 = x , e z^ = e x. The trigonometric functions become
cos z =
( e iz^ + e − iz ) = cos x cosh y − i sin x sinh y ,
sin z =
( e iz^ − e − iz ) = sin x cosh y + i cos x sinh y.
The hyperbolic functions become
cosh z =
( e z^ + e − z ) = cos iz , sinh z =
( e z^ − e − z ) = − i sin iz.
The natural logarithm generalizes to ln z = ln | z | + i (θ + 2 π n ), n = 0, ±1,... and general powers are defined as z p^ = e p^ ln^ z.
EXERCISES
6.1.1 (a) Find the reciprocal of x + iy , working entirely in the Cartesian representation. (b) Repeat part (a), working in polar form but expressing the final result in Cartesian form. 6.1.2 Prove algebraically that
| z 1 | − | z 2 | ≤ | z 1 + z 2 | ≤ | z 1 | + | z 2 |.
Interpret this result in terms of vectors. Prove that
| z − 1 | < |
z^2 − 1 | < | z + 1 |, for ( z ) > 0.
6.1.3 We may define a complex conjugation operator K such that Kz = z ∗. Show that K is not a linear operator. 6.1.4 Show that complex numbers have square roots and that the square roots are contained in the complex plane. What are the square roots of i?
330 Chapter 6 Functions of a Complex Variable I
6.1.9 Using the identities
cos z =
e iz^ + e − iz 2
, sin z =
e iz^ − e − iz 2 i
established from comparison of power series, show that (a) sin( x + iy ) = sin x cosh y + i cos x sinh y , cos( x + iy ) = cos x cosh y − i sin x sinh y , (b) |sin z |^2 = sin 2 x + sinh 2 y , |cos z |^2 = cos 2 x + sinh 2 y. This demonstrates that we may have |sin z |, |cos z | > 1 in the complex plane. 6.1.10 From the identities in Exercises 6.1.8 and 6.1.9, show that (a) sinh( x + iy ) = sinh x cos y + i cosh x sin y , cosh( x + iy ) = cosh x cos y + i sinh x sin y , (b) |sinh z |^2 = sinh 2 x + sin 2 y , |cosh z |^2 = sinh 2 x + cos 2 y.
6.1.11 Prove that (a) |sin z | ≥ |sin x | (b) |cos z | ≥ |cos x |. 6.1.12 Show that the exponential function e z^ is periodic with a pure imaginary period of 2π i. 6.1.13 Show that (a) tanh( z /2) =
sinh x + i sin y cosh x + cos y
, (b) coth( z /2) =
sinh x − i sin y cosh x − cos y
6.1.14 Find all the zeros of (a) sin z , (b) cos z , (c) sinh z , (d) cosh z. 6.1.15 Show that (a) sin−^1 z = − i ln( iz ±
1 − z^2 ), (d) sinh−^1 z = ln( z +
z^2 + 1), (b) cos−^1 z = − i ln( z ±
z^2 − 1), (e) cosh−^1 z = ln( z +
z^2 − 1),
(c) tan−^1 z =
i 2
ln
i + z i − z
, (f) tanh−^1 z =
ln
1 + z 1 − z
Hint. 1. Express the trigonometric and hyperbolic functions in terms of exponentials. 2. Solve for the exponential and then for the exponent. Note that sin−^1 z = arcsin z = (sin z )−^1 , etc. 6.1.16 A plane wave of light of angular frequency ω is represented by
e i ω( t − nx / c ).
In a certain substance the simple real index of refraction n is replaced by the complex quantity n − ik. What is the effect of k on the wave? What does k correspond to physically? The generalization of a quantity from real to complex form occurs frequently in physics. Examples range from the complex Young’s modulus of viscoelastic materials to the complex (optical) potential of the “cloudy crystal ball” model of the
6.2 Cauchy--Riemann Conditions 331
atomic nucleus. See the chapter on the optical model in M. A. Preston, Structure of the Nucleus. Addison-Wesley, Reading, MA (1993). 6.1.17 A damped simple harmonic oscillator is driven by the complex external force Fe i ω t^. Show that the steady-state amplitude is given by,
A =
F
m
ω^20 − ω^2
Explain the resonance condition and relate m , ω 0 , b to the oscillator parameters. Hint. Find a complex solution z ( t ) = Ae i ω t^ of the ordinary differential equation. 6.1.18 We see that for the angular momentum components defined in Exercise 2.5.11, L (^) x − iL (^) y = ( L (^) x + iL (^) y )∗. Explain why this happens. 6.1.19 Show that the phase of f ( z ) = u + i v is equal to the imaginary part of the logarithm of f ( z ). Exercise 10.2.13 depends on this result. 6.1.20 (a) Show that e ln^ z^ always equals z. (b) Show that ln e z^ does not always equal z. 6.1.21 Verify the consistency of the combination rules of impedances with those of resistances, inductances, and capacitances by considering circuits with resistors only, etc. Derive the combination rules from Kirchhoff’s laws. Describe the origin of Kirchhoff’s laws. 6.1.22 Show that negative numbers have logarithms in the complex plane. In particular, find ln(−1). ANS. ln(−1) = i π.
6.2 Cauchy--Riemann Conditions
Having established complex functions of a complex variable, we now proceed to differentiate them. The derivative of f ( z ) = u ( x , y ) + i v( x , y ), like that of a real function, is defined by
lim δ z → 0
f ( z + δ z ) − f ( z ) z + δ z − z
= lim δ z → 0
δ f ( z ) δ z
d f dz
= f ′( z ), (6.22)
provided that the limit is independent of the particular approach to the point z. For real variables we require that the right-hand limit ( x → x 0 from above) and the left-hand limit ( x → x 0 from below) be equal for the derivative d f ( x )/ dx to exist at x = x 0. Now, with z (or z 0 ) some point in a plane, our requirement that the limit be independent of the direction of approach is very restrictive. Consider increments δ x and δ y of the variables x and y , respectively. Then δ z = δ x + i δ y. (6.23)
6.2 Cauchy--Riemann Conditions 333
v ( x, y ) = const. u ( x, y ) = const.
Figure 6.
Orthogonal Tangents to u ( x , y ) = Const. v ( x , y ) = Const. Lines
Now recall the geometric meaning of − ux / uy as the slope of the tangent [see Eq. (1.54)] of each curve u ( x , y ) = const. and similarly for v( x , y ) = const. (Fig. 6.6). Thus, Eq. (6.29) means that the u = const. and v = const. curves are mutually orthogonal at each intersection because sin β = sin(α + 90 ◦) = cos α and cos β = − sin α imply tan β · tan α = −1 by taking the ratio. Alternatively,
ux dx + uy dy = 0 = v y dx − v x dy
state that if ( dx , dy ) is tangent to the u -curve, then the orthogonal (− dy , dx ) is tangent to the v-curve at the intersection point z = ( x , y ). Equivalently, ux v x + uy v y = 0 implies that the gradient vectors ( ux , uy ) and (v x , v y ) are perpendicular. Conversely, if the Cauchy–Riemann conditions are satisfied and the partial derivatives of u ( x , y ) and v( x , y ) are continuous, the derivative d f / dz exists. This may be shown by writing
δ f =
∂ u ∂ x
∂v ∂ x
δ x +
∂ u ∂ y
∂v ∂ y
δ y. (6.30)
The justification for this expression depends on the continuity of the partial derivatives of u and v. Dividing by δ z , we have
δ f δ z
(∂ u /∂ x + i (∂v/∂ x ))δ x + (∂ u /∂ y + i (∂ u /∂ y ))δ y δ x + i δ y
=
(∂ u /∂ x + i (∂v/∂ x )) + (∂ u /∂ y + i (∂v/∂ y ))δ y /δ x 1 + i (δ y /δ x )
If δ f /δ z is to have a unique value, the dependence on δ y /δ x must be elimi- nated. Applying the Cauchy–Riemann conditions to the y derivatives, we obtain
∂ u ∂ y
∂v ∂ y
∂v ∂ x
∂ u ∂ x
Substituting Eq. (6.32) into Eq. (6.30), we may rewrite the δ y and δ x depen- dence as δ z = δ x + i δ y and obtain
δ f δ z
∂ u ∂ x
∂v ∂ x
which shows that lim δ f /δ z is independent of the direction of approach in the complex plane as long as the partial derivatives are continuous.
334 Chapter 6 Functions of a Complex Variable I
It is worthwhile to note that the Cauchy–Riemann conditions guarantee that the curves u = c 1 = constant will be orthogonal to the curves v = c 2 = constant (compare Section 2.1). This property is fundamental in application to potential problems in a variety of areas of physics. If u = c 1 is a line of electric force, then v = c 2 is an equipotential line (surface) and vice versa. Also, it is easy to show from Eq. (6.28) that both u and v satisfy Laplace’s equation. A further implication for potential theory is developed in Exercise 6.2.2. We have already generalized the elementary functions to the complex plane by replacing the real variable x by complex z. Let us now check that their derivatives are the familiar ones.
EXAMPLE 6.2.1 Derivatives of Elementary Functions^ We define the elementary functions by their Taylor expansions (see Section 5.6, with x → z , and Section 6.5)
e z^ =
n = 0
zn n!
sin z =
n = 0
(−1) n^
z^2 n +^1 (2 n + 1)!
, cos z =
n = 0
(−1) n^
z^2 n (2 n )!
ln(1 + z ) =
n = 1
(−1) n −^1
zn n
We differentiate termwise [which is justified by absolute convergence for e z , cos z , sin z for all z and for ln(1 + z ) for | z | < 1] and see that
d dz
zn^ = lim δ z → 0
( z + δ z ) n^ − zn δ z
= lim δ z → 0 [ zn^ + nzn −^1 δ z + · · · + (δ z ) n^ − zn ]/δ z = nz n −^1 ,
de z dz
n = 1
nz n −^1 n!
= e z ,
d sin z dz
n = 0
(−1) n^
(2 n + 1) z^2 n (2 n + 1)!
= cos z ,
d cos z dz
n = 1
(−1) n^
2 n z^2 n −^1 (2 n )!
= − sin z ,
d ln(1 + z ) dz
d dz
∑^ ∞
n = 1
(−1) n −^1
zn n
n = 1
(−1) n −^1 zn −^1 =
1 + z
that is, the real derivative results all generalize to the complex field, simply replacing x → z. ■
336 Chapter 6 Functions of a Complex Variable I
EXAMPLE 6.2.3 Let^ f^ ( z )^ =^ z ∗. Now u = x and v = − y. Applying the Cauchy–Riemann condi- tions, we obtain ∂ u ∂ x
= 1, whereas
∂v ∂ y
The Cauchy–Riemann conditions are not satisfied and f ( z ) = z ∗^ is not an analytic function of z. It is interesting to note that f ( z ) = z ∗^ is continuous, thus providing an example of a function that is everywhere continuous, but nowhere differentiable. ■
The derivative of a real function of a real variable is essentially a local characteristic in that it provides information about the function only in a local neighborhood—for instance, as a truncated Taylor expansion. The existence of a derivative of a function of a complex variable has much more far-reaching implications. The real and imaginary parts of analytic functions must sepa- rately satisfy Laplace’s equation. This is Exercise 6.2.1. Furthermore, an ana- lytic function is guaranteed derivatives of all orders (Section 6.4). In this sense the derivative not only governs the local behavior of the complex function but also controls the distant behavior.
EXERCISES
6.2.1 The functions u ( x , y ) and v( x , y ) are the real and imaginary parts, re- spectively, of an analytic function w( z ). (a) Assuming that the required derivatives exist, show that ∇^2 u = ∇^2 v = 0. Solutions of Laplace’s equation, such as u ( x , y ) and v( x , y ), are called harmonic functions. (b) Show that ∂ u ∂ x
∂ u ∂ y
∂v ∂ x
∂v ∂ y
and give a geometric interpretation. Hint. The technique of Section 1.6 allows you to construct vectors nor- mal to the curve u ( x , y ) = ci and v( x , y ) = c (^) j. 6.2.2 Show whether or not the function f ( z ) = ( z ) = x is analytic. 6.2.3 Having shown that the real part u ( x , y ) and the imaginary part v( x , y ) of an analytic function w( z ) each satisfy Laplace’s equation, show that u ( x , y ) and v( x , y ) cannot both have either a maximum or a mini- mum in the interior of any region in which w( z ) is analytic. (They can have saddle points ; see Section 7.4.) 6.2.4 Let A = ∂^2 w/∂ x^2 , B = ∂^2 w/∂ x ∂ y , C = ∂^2 w/∂ y^2. From the calculus of functions of two variables, w( x , y ), we have a saddle point if B^2 − AC > 0.
6.3 Cauchy’s Integral Theorem 337
With f ( z ) = u ( x , y ) + i v( x , y ), apply the Cauchy–Riemann conditions and show that both u ( x , y ) and v( x , y ) do not have a maximum or a minimum in a finite region of the complex plane. (See also Section 7.4.) 6.2.5 Find the analytic function w( z ) = u ( x , y ) + i v( x , y ) if (a) u ( x , y ) = x^3 − 3 xy^2 , (b) v( x , y ) = e − y^ sin x. 6.2.6 If there is some common region in which w 1 = u ( x , y ) + i v( x , y ) and w 2 = w 1 ∗ = u ( x , y ) − i v( x , y ) are both analytic, prove that u ( x , y ) and v( x , y ) are constants. 6.2.7 The function f ( z ) = u ( x , y ) + i v( x , y ) is analytic. Show that f ∗( z ∗) is also analytic. 6.2.8 A proof of the Schwarz inequality (Section 9.4) involves minimizing an expression f = ψ aa + λψ ab + λ∗ψ ab ∗ + λλ∗ψ bb ≥ 0. The ψ are integrals of products of functions; ψ aa and ψ bb are real, ψ ab is complex, and λ is a complex parameter. (a) Differentiate the preceding expression with respect to λ∗, treating λ as an independent parameter, independent of λ∗. Show that setting the derivative ∂ f /∂λ∗^ equal to zero yields λ = −ψ ab ∗/ψ bb. (b) Show that ∂ f /∂λ = 0 leads to the same result. (c) Let λ = x + iy , λ∗^ = x − iy. Set the x and y derivatives equal to zero and show that again λ = ψ ab ∗/ψ bb. 6.2.9 The function f ( z ) is analytic. Show that the derivative of f ( z ) with re- spect to z ∗^ does not exist unless f ( z ) is a constant. Hint. Use the chain rule and take x = ( z + z ∗)/2, y = ( z − z ∗)/ 2 i. Note. This result emphasizes that our analytic function f ( z ) is not just a complex function of two real variables x and y. It is a function of the complex variable x + iy.
6.3 Cauchy’s Integral Theorem
Contour Integrals
With differentiation under control, we turn to integration. The integral of a complex variable over a contour in the complex plane may be defined in close analogy to the (Riemann) integral of a real function integrated along the real x -axis and line integrals of vectors in Chapter 1. The contour integral may be
