CHAPTER 13

Complex Numbers and Functions. Complex Differentiation

The transition from “real calculus” to “complex calculus” starts with a discussion of complex numbers and their geometric representation in the complex plane. We then progress to analytic functions in Sec. 13.3. We desire functions to be analytic because these are the “useful functions” in the sense that they are differentiable in some domain and operations of complex analysis can be applied to them. The most important equations are therefore the Cauchy–Riemann equations in Sec. 13.4 because they allow a test of analyticity of such functions. Moreover, we show how the Cauchy–Riemann equations are related to the important Laplace equation.

The remaining sections of the chapter are devoted to elementary complex functions (exponential, trigonometric, hyperbolic, and logarithmic functions). These generalize the familiar real functions of calculus. Detailed knowledge of them is an absolute necessity in practical work, just as that of their real counterparts is in calculus.

Prerequisite: Elementary calculus.

References and Answers to Problems: App. 1 Part D, App. 2.

13.1 Complex Numbers and Their Geometric Representation

The material in this section will most likely be familiar to the student and serve as a review.

Equations without real solutions, such as x² = −1 or x² − 10x + 40 = 0, were observed early in history and led to the introduction of complex numbers.¹ By definition, a complex number z is an ordered pair (x, y) of real numbers x and y, written

x is called the real part and y the imaginary part of z, written

By definition, two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

(0, 1) is called the imaginary unit and is denoted by i,

Addition, Multiplication. Notation z = x + iy

Addition of two complex numbers z₁ = (x₁, y₁) and z₂ = (x₂,y₂) is defined by

Multiplication is defined by

These two definitions imply that

and

as for real numbers x₁, x₂. Hence the complex numbers “extend” the real numbers. We can thus write

because by (1), and the definition of multiplication, we have

Together we have, by addition, (x, y) = (x, 0) + (0, y) = x + iy.

In practice, complex numbers z = (x, y) are written

or z = x + yi, e.g., 17 + 4i (instead of i4).

Electrical engineers often write j instead of i because they need i for the current. If x = 0, then z = iy and is called pure imaginary. Also, (1) and (3) give

because, by the definition of multiplication, i² = ii = (0, 1)(0, 1) = (−1, 0) = −1.

For addition the standard notation (4) gives [see (2)]

For multiplication the standard notation gives the following very simple recipe. Multiply each term by each other term and use i² = −1 when it occurs [see (3)]:

This agrees with (3). And it shows that x + iy is a more practical notation for complex numbers than (x, y).

If you know vectors, you see that (2) is vector addition, whereas the multiplication (3) has no counterpart in the usual vector algebra.

Subtraction, Division

Subtraction and division are defined as the inverse operations of addition and multiplication, respectively. Thus the difference z = z₁ − z₂ is the complex number z for which z₁ = z + z₂. Hence by (2),

The quotient z = z₁/z₂ (z₂ ≠ 0) is the complex number z for which z₁ = zz₂. If we equate the real and the imaginary parts on both sides of this equation, setting z = x + iy, we obtain x₁ = x₂x − y₂y, y₁ = y₂x + x₂y. The solution is

The practical rule used to get this is by multiplying numerator and denominator of z₁/z₂ by x₂ − iy₂ and simplifying:

Complex numbers satisfy the same commutative, associative, and distributive laws as real numbers (see the problem set).

Complex Plane

So far we discussed the algebraic manipulation of complex numbers. Consider the geometric representation of complex numbers, which is of great practical importance. We choose two perpendicular coordinate axes, the horizontal x-axis, called the real axis, and the vertical y-axis, called the imaginary axis. On both axes we choose the same unit of length (Fig. 318). This is called a Cartesian coordinate system.

Fig. 318. The complex plane

Fig. 319. The number 4 − 3i in the complex plane

We now plot a given complex number z = (x, y) = x + iy as the point P with coordinates x, y. The xy-plane in which the complex numbers are represented in this way is called the complex plane.² Figure 319 shows an example.

Instead of saying “the point represented by z in the complex plane” we say briefly and simply “the point z in the complex plane.” This will cause no misunderstanding.

Addition and subtraction can now be visualized as illustrated in Figs. 320 and 321.

Fig. 320. Addition of complex numbers

Fig. 321. Subtraction of complex numbers

Complex Conjugate Numbers

The complex conjugate of a complex number z = x + iy is defined by

It is obtained geometrically by reflecting the point z in the real axis. Figure 322 shows this for z = 5 + 2i and its conjugate = 5 − 2i.

Fig. 322. Complex conjugate numbers

The complex conjugate is important because it permits us to switch from complex to real. Indeed, by multiplication, z = x² + y² (verify!). By addition and subtraction, z + = 2x, z − = 2iy. We thus obtain for the real part x and the imaginary part y (not iy!) of z =x + iy the important formulas

If z is real, z = x, then = z by the definition of , and conversely. Working with conjugates is easy, since we have

13.2 Polar Form of Complex Numbers. Powers and Roots

We gain further insight into the arithmetic operations of complex numbers if, in addition to the xy-coordinates in the complex plane, we also employ the usual polar coordinates r, θ defined by

We see that then z = x + iy takes the so-called polar form

r is called the absolute value or modulus of z and is denoted by |z|. Hence

Geometrically, |z| is the distance of the point z from the origin (Fig. 323). Similarly, |z₁ − z₂| is the distance between z₁ and z₂ (Fig. 324).

θ is called the argument of z and is denoted by arg z. Thus θ = arg z and (Fig. 323)

Geometrically, θ is the directed angle from the positive x-axis to OP in Fig. 323. Here, as in calculus, all angles are measured in radians and positive in the counterclockwise sense.

For z = 0 this angle θ is undefined. (Why?) For a given z ≠ 0 it is determined only up to integer multiples of 2π since cosine and sine are periodic with period 2π. But one often wants to specify a unique value of arg z of a given z ≠ 0. For this reason one defines the principal value Arg z (with capital A!) of arg z by the double inequality

Then we have Arg z = 0 for positive real z = x, which is practical, and Arg z = π (not −π!) for negative real z, e.g., for z = −4. The principal value (5) will be important in connection with roots, the complex logarithm (Sec. 13.7), and certain integrals. Obviously, for a given z ≠ 0, the other values of arg z are arg z = Arg z ± 2nπ(n = ±1, ±2, …).

Fig. 323. Complex plane, polar form of a complex number

Fig. 324. Distance between two points in the complex plane

Triangle Inequality

Inequalities such as x₁ < x₂ make sense for real numbers, but not in complex because there is no natural way of ordering complex numbers. However, inequalities between absolute values (which are real!), such as |z₁| < |z₂ (meaning that z₁ is closer to the origin than z₂) are of great importance. The daily bread of the complex analyst is the triangle inequality

which we shall use quite frequently. This inequality follows by noting that the three points 0, z₁ and z₁ + z₂ are the vertices of a triangle (Fig. 326) with sides |z₁|, |z₂|, and |z₁ + z₂|, and one side cannot exceed the sum of the other two sides. A formal proof is left to the reader (Prob. 33). (The triangle degenerates if z₁ and z₂ lie on the same straight line through the origin.)

Fig. 326. Triangle inequality

By induction we obtain from (6) the generalized triangle inequality

that is, the absolute value of a sum cannot exceed the sum of the absolute values of the terms.

Multiplication and Division in Polar Form

This will give us a “geometrical” understanding of multiplication and division. Let

Multiplication. By (3) in Sec. 13.1 the product is at first

The addition rules for the sine and cosine [(6) in App. A3.1] now yield

Taking absolute values on both sides of (7), we see that the absolute value of a product equals the product of the absolute values of the factors,

Taking arguments in (7) shows that the argument of a product equals the sum of the arguments of the factors,

Division. We have z₁ = (z₁/z₂)z₂. Hence |z₁| = |(z₁/z₂)z₂| = |z₁/z₂||z₂| and by division by |z₂|

Similarly, arg z₁ = arg [(z₁/z₂) z2] = arg (z₁/z₂) + arg z₂ and by subtraction of arg z₂

Combining (10) and (11) we also have the analog of (7),

To comprehend this formula, note that it is the polar form of a complex number of absolute value r₁/r₂ and argument θ₁ − θ₂. But these are the absolute value and argument of z₁/z₂, as we can see from (10), (11), and the polar forms of z₁ and z₂.

Roots

If z = wⁿ (n = 1, 2, …), then to each value of w there corresponds one value of z. We shall immediately see that, conversely, to a given z ≠ 0 there correspond precisely n distinct values of w. Each of these values is called an nth root of z, and we write

Hence this symbol is multivalued, namely, n-valued. The n values of can be obtained as follows. We write z and w in polar form

Then the equation wⁿ = z becomes, by De Moivre's formula (with ø instead of θ),

The absolute values on both sides must be equal; thus, Rⁿ = r, so that where is positive real (an absolute value must be nonnegative!) and thus uniquely determined. Equating the arguments nø and θ and recalling that θ is determined only up to integer multiples of 2π, we obtain

where k is an integer. For k = 0, 1, …, n − 1 we get n distinct values of w. Further integers of k would give values already obtained. For instance, k = n gives 2kπ/n = 2π, hence the w corresponding to k = 0, etc. Consequently, , for z ≠ 0, has the n distinct values

where k = 0, 1, …, n − 1. These n values lie on a circle of radius with center at the origin and constitute the vertices of a regular polygon of n sides. The value of obtained by taking the principal value of arg z and k = 0 in (15) is called the principal value of

Taking z = 1 in (15), we have |z| = r = 1 and Arg z = 0. Then (15) gives

These n values are called the nth roots of unity. They lie on the circle of radius 1 and center 0, briefly called the unit circle (and used quite frequently!). Figures 327–329 show

Fig. 327.

Fig. 328.

Fig. 329.

If ω denotes the value corresponding to k = 1 in (16), then the n values of can be written as

More generally, if w₁ is any nth root of an arbitrary complex number z (≠ 0), then the n values of in (15) are

because multiplying w₁ by ω^k corresponds to increasing the argument of w₁ by 2kπ/n. Formula (17) motivates the introduction of roots of unity and shows their usefulness.

13.3 Derivative. Analytic Function

Just as the study of calculus or real analysis required concepts such as domain, neighborhood, function, limit, continuity, derivative, etc., so does the study of complex analysis. Since the functions live in the complex plane, the concepts are slightly more difficult or different from those in real analysis. This section can be seen as a reference section where many of the concepts needed for the rest of Part D are introduced.

Circles and Disks. Half-Planes

The unit circle |z| = 1 (Fig. 330) has already occurred in Sec. 13.2. Figure 331 shows a general circle of radius ρ and center a. Its equation is

Fig. 330. Unit circle

Fig. 331. Circle in the complex plane

Fig. 332. Annulus in the complex plane

because it is the set of all z whose distance |z − a| from the center a equals ρ. Accordingly, its interior (“open circular disk”) is given by |z − a| < ρ, its interior plus the circle itself (“closed circular disk”) by |z − a| ρ, and its exterior by |z − a| > ρ. As an example, sketch this for a = 1 + i and ρ = 2, to make sure that you understand these inequalities.

An open circular disk |z − a| < ρ is also called a neighborhood of a or, more precisely, a ρ-neighborhood of a. And a has infinitely many of them, one for each value of ρ(>0), and a is a point of each of them, by definition!

In modern literature any set containing a ρ-neighborhood of a is also called a neighborhood of a.

Figure 332 shows an open annulus (circular ring) ρ₁ < |z − a| <ρ₂, which we shall need later. This is the set of all z whose distance |z − a| from a is greater than ρ₁ but less than ρ₂. Similarly, the closed annulus ρ₁ |z − a| ρ₂ includes the two circles.

Half-Planes. By the (open) upper half-plane we mean the set of all points z = x + iy such that y > 0. Similarly, the condition y < 0 defines the lower half-plane, x > 0 the right half-plane, and x < 0 the left half-plane.

For Reference: Concepts on Sets in the Complex Plane

To our discussion of special sets let us add some general concepts related to sets that we shall need throughout Chaps. 13–18; keep in mind that you can find them here.

By a point set in the complex plane we mean any sort of collection of finitely many or infinitely many points. Examples are the solutions of a quadratic equation, the points of a line, the points in the interior of a circle as well as the sets discussed just before.

A set S is called open if every point of S has a neighborhood consisting entirely of points that belong to S. For example, the points in the interior of a circle or a square form an open set, and so do the points of the right half-plane Re z = x > 0.

A set S is called connected if any two of its points can be joined by a chain of finitely many straight-line segments all of whose points belong to S. An open and connected set is called a domain. Thus an open disk and an open annulus are domains. An open square with a diagonal removed is not a domain since this set is not connected. (Why?)

The complement of a set S in the complex plane is the set of all points of the complex plane that do not belong to S. A set S is called closed if its complement is open. For example, the points on and inside the unit circle form a closed set (“closed unit disk”) since its complement |z| > 1 is open.

A boundary point of a set S is a point every neighborhood of which contains both points that belong to S and points that do not belong to S. For example, the boundary points of an annulus are the points on the two bounding circles. Clearly, if a set S is open, then no boundary point belongs to S; if S is closed, then every boundary point belongs to S. The set of all boundary points of a set S is called the boundary of S.

A region is a set consisting of a domain plus, perhaps, some or all of its boundary points. WARNING! “Domain” is the modern term for an open connected set. Nevertheless, some authors still call a domain a “region” and others make no distinction between the two terms.

Complex Function

Complex analysis is concerned with complex functions that are differentiable in some domain. Hence we should first say what we mean by a complex function and then define the concepts of limit and derivative in complex. This discussion will be similar to that in calculus. Nevertheless it needs great attention because it will show interesting basic differences between real and complex calculus.

Recall from calculus that a real function f defined on a set S of real numbers (usually an interval) is a rule that assigns to every x in S a real number f(x), called the value of f at x. Now in complex, S is a set of complex numbers. And a function f defined on S is a rule that assigns to every z in S a complex number w, called the value of f at z. We write

Here z varies in S and is called a complex variable. The set S is called the domain of definition of f or, briefly, the domain of f. (In most cases S will be open and connected, thus a domain as defined just before.)

Example: w = f(z) = z² + 3z is a complex function defined for all z; that is, its domain S is the whole complex plane.

The set of all values of a function f is called the range of f.

w is complex, and we write w = u + iv, where u and v are the real and imaginary parts, respectively. Now w depends on z = x + iy. Hence u becomes a real function of x and y, and so does v. We may thus write

This shows that a complex function f(z) is equivalent to a pair of real functions u(x, y) and v(x, y), each depending on the two real variables x and y.

Remarks on Notation and Terminology

1. Strictly speaking, f(z) denotes the value of f at z, but it is a convenient abuse of language to talk about the function f(z) (instead of the function f), thereby exhibiting the notation for the independent variable.

2. We assume all functions to be single-valued relations, as usual: to each z in S there corresponds but one value w = f(z) (but, of course, several z may give the same value w = f(z), just as in calculus). Accordingly, we shall not use the term “multivalued function” (used in some books on complex analysis) for a multivalued relation, in which to a z there corresponds more than one w.

Limit, Continuity

A function f(z) is said to have the limit l as z approaches a point z₀, written

if f is defined in a neighborhood of z₀ (except perhaps at z₀ itself) and if the values of f are “close” to l for all z “close” to z₀; in precise terms, if for every positive real we can find a positive real δ such that for all z ≠ z₀ in the disk |z − z₀| < δ (Fig. 333) we have

geometrically, if for every z ≠ z₀ in that δ-disk the value of f lies in the disk (2).

Formally, this definition is similar to that in calculus, but there is a big difference. Whereas in the real case, x can approach an x₀ only along the real line, here, by definition, z may approach z₀ from any direction in the complex plane. This will be quite essential in what follows.

If a limit exists, it is unique. (See Team Project 24.)

A function f(z) is said to be continuous at z = z₀ if f(z₀) is defined and

Note that by definition of a limit this implies that f(z) is defined in some neighborhood of z₀.

f(z) is said to be continuous in a domain if it is continuous at each point of this domain.

Fig. 333. Limit

Derivative

The derivative of a complex function f at a point z₀ is written f′(z₀) and is defined by

provided this limit exists. Then f is said to be differentiable at z₀. If we write Δz = z − z₀, we have z = z₀ + Δz and (4) takes the form

Now comes an important point. Remember that, by the definition of limit, f(z) is defined in a neighborhood of z₀ and z in (4′) may approach z₀ from any direction in the complex plane. Hence differentiability at z₀ means that, along whatever path z approaches z₀, the quotient in (4′) always approaches a certain value and all these values are equal. This is important and should be kept in mind.

The differentiation rules are the same as in real calculus, since their proofs are literally the same. Thus for any differentiable functions f and g and constant c we have

as well as the chain rule and the power rule (zⁿ)′ = nzⁿ⁻¹(n integer).

Also, if f(z) is differentiable at z₀, it is continuous at z₀. (See Team Project 24.)

Fig. 334. Paths in (5)

Surprising as Example 4 may be, it merely illustrates that differentiability of a complex function is a rather severe requirement.

The idea of proof (approach of z from different directions) is basic and will be used again as the crucial argument in the next section.

Analytic Functions

Complex analysis is concerned with the theory and application of “analytic functions,” that is, functions that are differentiable in some domain, so that we can do “calculus in complex.” The definition is as follows.

Hence analyticity of f(z) at z₀ means that f(z) has a derivative at every point in some neighborhood of z₀ (including z₀ itself since, by definition, z₀ is a point of all its neighborhoods). This concept is motivated by the fact that it is of no practical interest if a function is differentiable merely at a single point z₀ but not throughout some neighborhood of z₀. Team Project 24 gives an example.

A more modern term for analytic in D is holomorphic in D.

The concepts discussed in this section extend familiar concepts of calculus. Most important is the concept of an analytic function, the exclusive concern of complex analysis. Although many simple functions are not analytic, the large variety of remaining functions will yield a most beautiful branch of mathematics that is very useful in engineering and physics.

13.4 Cauchy–Riemann Equations. Laplace's Equation

As we saw in the last section, to do complex analysis (i.e., “calculus in the complex”) on any complex function, we require that function to be analytic on some domain that is differentiable in that domain.

The Cauchy–Riemann equations are the most important equations in this chapter and one of the pillars on which complex analysis rests. They provide a criterion (a test) for the analyticity of a complex function

Roughly, f is analytic in a domain D if and only if the first partial derivatives of u and v satisfy the two Cauchy–Riemann equations⁴

everywhere in D; here u_x = ∂u/∂y and u_y = ∂u/∂y (and similarly for v) are the usual notations for partial derivatives. The precise formulation of this statement is given in Theorems 1 and 2.

Example: f(z) = z² = x² − y² + 2ixy is analytic for all z (see Example 3 in Sec. 13.3), and u = x² − y² and v = 2xy satisfy (1), namely, u_x = 2x = v_y as well as u_y = −2y = −v_x. More examples will follow.

Formulas (4) and (5) are also quite practical for calculating derivatives f′(z) as we shall see.

The Cauchy–Riemann equations are fundamental because they are not only necessary but also sufficient for a function to be analytic. More precisely, the following theorem holds.

The proof is more involved than that of Theorem 1 and we leave it optional (see App. 4).

Theorems 1 and 2 are of great practical importance, since, by using the Cauchy–Riemann equations, we can now easily find out whether or not a given complex function is analytic.

We mention that, if we use the polar form z = r(cos θ + i sin θ) and set f(z) = u(r, θ) + iv(r, θ), then the Cauchy–Riemann equations are (Prob. 1)

Laplace's Equation. Harmonic Functions

The great importance of complex analysis in engineering mathematics results mainly from the fact that both the real part and the imaginary part of an analytic function satisfy Laplace's equation, the most important PDE of physics. It occurs in gravitation, electrostatics, fluid flow, heat conduction, and other applications (see Chaps. 12 and 18).

Solutions of Laplace's equation having continuous second-order partial derivatives are called harmonic functions and their theory is called potential theory (see also Sec. 12.11). Hence the real and imaginary parts of an analytic function are harmonic functions.

If two harmonic functions u and v satisfy the Cauchy–Riemann equations in a domain D, they are the real and imaginary parts of an analytic function f in D. Then v is said to be a harmonic conjugate function of u in D. (Of course, this has absolutely nothing to do with the use of “conjugate” for .)

Example 4 illustrates that a conjugate of a given harmonic function is uniquely determined up to an arbitrary real additive constant.

The Cauchy–Riemann equations are the most important equations in this chapter. Their relation to Laplace's equation opens a wide range of engineering and physical applications, as shown in Chap. 18.

13.5 Exponential Function

In the remaining sections of this chapter we discuss the basic elementary complex functions, the exponential function, trigonometric functions, logarithm, and so on. They will be counterparts to the familiar functions of calculus, to which they reduce when z = x is real. They are indispensable throughout applications, and some of them have interesting properties not shared by their real counterparts.

We begin with one of the most important analytic functions, the complex exponential function

The definition of e^x in terms of the real functions e^x, cos y, and sin y is

This definition is motivated by the fact the e^z extends the real exponential function e^x of calculus in a natural fashion. Namely:

(A) e^z = e^x for real z = x because cos y = 1 and sin y = 0 when y = 0.

(B) e^z is analytic for all z. (Proved in Example 2 of Sec. 13.4.)

(C) The derivative of e^z is e^z, that is,

This follows from (4) in Sec. 13.4,

Further Properties. A function f(z) that is analytic for all z is called an entire function. Thus, e^z is entire. Just as in calculus the functional relation

holds for any and z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ Indeed, by (1),

Since e^x1e^x2 = e^x1+x2 for these real functions, by an application of the addition formulas for the cosine and sine functions (similar to that in Sec. 13.2) we see that

as asserted. An interesting special case of (3) is z₁ = x, z₂ = iy; then

Furthermore, for z = iy we have from (1) the so-called Euler formula

Hence the polar form of a complex number, z = r(cos θ + i sin θ), may now be written

From (5) we obtain

as well as the important formulas (verify!)

Another consequence of (5) is

That is, for pure imaginary exponents, the exponential function has absolute value 1, a result you should remember. From (9) and (1),

since |e^z| = e^x shows that (1) is actually e^z in polar form.

From |e^z| = e^x ≠ 0 in (10) we see that

So here we have an entire function that never vanishes, in contrast to (nonconstant) polynomials, which are also entire (Example 5 in Sec. 13.3) but always have a zero, as is proved in algebra.

Periodicity of e^x with period 2πi,

is a basic property that follows from (1) and the periodicity of cos y and sin y. Hence all the values that w = e^z can assume are already assumed in the horizontal strip of width 2π

This infinite strip is called a fundamental region of e^z.

To summarize: many properties of e^z = exp z parallel those of e^x; an exception is the periodicity of e^z with 2πi, which suggested the concept of a fundamental region. Keep in mind that e^z is an entire function. (Do you still remember what that means?)

Fig. 336. Fundamental region of the exponential function e^z in the z-plane

13.6 Trigonometric and Hyperbolic Functions. Euler's Formula

Just as we extended the real e^x to the complex e^z in Sec. 13.5, we now want to extend the familiar real trigonometric functions to complex trigonometric functions. We can do this by the use of the Euler formulas (Sec. 13.5)

By addition and subtraction we obtain for the real cosine and sine

This suggests the following definitions for complex values z = x + iy:

It is quite remarkable that here in complex, functions come together that are unrelated in real. This is not an isolated incident but is typical of the general situation and shows the advantage of working in complex.

Furthermore, as in calculus we define

and

Since e^z is entire, cos z and sin z are entire functions. tan z and sec z are not entire; they are analytic except at the points where cos z is zero; and cot z and csc z are analytic except where sin z is zero. Formulas for the derivatives follow readily from (e^z)′ = e^z and (1)–(3); as in calculus,

etc. Equation (1) also shows that Euler's formula is valid in complex:

The real and imaginary parts of cos z and sin z are needed in computing values, and they also help in displaying properties of our functions. We illustrate this with a typical example.

General formulas for the real trigonometric functions continue to hold for complex values. This follows immediately from the definitions. We mention in particular the addition rules

and the formula

Some further useful formulas are included in the problem set.

Hyperbolic Functions

The complex hyperbolic cosine and sine are defined by the formulas

This is suggested by the familiar definitions for a real variable [see (8)]. These functions are entire, with derivatives

as in calculus. The other hyperbolic functions are defined by

Complex Trigonometric and Hyperbolic Functions Are Related. If in (11), we replace z by iz and then use (1), we obtain

Similarly, if in (1) we replace z by iz and then use (11), we obtain conversely

Here we have another case of unrelated real functions that have related complex analogs, pointing again to the advantage of working in complex in order to get both a more unified formalism and a deeper understanding of special functions. This is one of the main reasons for the importance of complex analysis to the engineer and physicist.

13.7 Logarithm. General Power. Principal Value

We finally introduce the complex logarithm, which is more complicated than the real logarithm (which it includes as a special case) and historically puzzled mathematicians for some time (so if you first get puzzled—which need not happen!—be patient and work through this section with extra care).

The natural logarithm of z = x + iy is denoted by ln z (sometimes also by log z) and is defined as the inverse of the exponential function; that is, w = ln z is defined for z ≠ 0 by the relation

(Note that z = 0 is impossible, since e^w ≠ 0 for all w; see Sec. 13.5.) If we set w = u + iv and z = re^iθ, this becomes

Now, from Sec. 13.5, we know that e^u+iv has the absolute value e^u and the argument v. These must be equal to the absolute value and argument on the right:

e^u = r gives u = ln r, where ln r is the familiar real natural logarithm of the positive number r = |z|. Hence w = u + iv = ln z is given by

Now comes an important point (without analog in real calculus). Since the argument of z is determined only up to integer multiples of 2π, the complex natural logarithm ln z(z ≠ 0) is infinitely many-valued.

The value of ln z corresponding to the principal value Arg z (see Sec. 13.2) is denoted by Ln z (Ln with capital L) and is called the principal value of ln z. Thus

The uniqueness of Arg z for given z (≠ 0) implies that Ln z is single-valued, that is, a function in the usual sense. Since the other values of arg z differ by integer multiples of 2π, the other values of ln z are given by

They all have the same real part, and their imaginary parts differ by integer multiples of 2π.

If z is positive real, then Arg z = 0, and Ln z becomes identical with the real natural logarithm known from calculus. If z is negative real (so that the natural logarithm of calculus is not defined!), then Arg z = π and

From (1) and e^{ln r} = r for positive real r we obtain

as expected, but since arg (e^z) = y ± 2nπ is multivalued, so is

Fig. 337. Some values of ln (3 − 4i) in Example 1

The familiar relations for the natural logarithm continue to hold for complex values, that is,

but these relations are to be understood in the sense that each value of one side is also contained among the values of the other side; see the next example.

Each of the infinitely many functions in (3) is called a branch of the logarithm. The negative real axis is known as a branch cut and is usually graphed as shown in Fig. 338. The branch for n = 0 is called the principal branch of ln z.

Fig. 338. Branch cut for ln z

General Powers

General powers of a complex number z = x + iy are defined by the formula

Since ln z is infinitely many-valued, z^c will, in general, be multivalued. The particular value

is called the principal value of z^c.

If c = n = 1, 2, …, then zⁿ is single-valued and identical with the usual nth power of z.

If c = −1, −2, …, the situation is similar.

If c = 1/n, where n = 2, 3, …, then

the exponent is determined up to multiples of 2πi/n and we obtain the n distinct values of the nth root, in agreement with the result in Sec. 13.2. If c = p/q, the quotient of two positive integers, the situation is similar, and z^c has only finitely many distinct values. However, if c is real irrational or genuinely complex, then z^c is infinitely many-valued.

It is a convention that for real positive z = x the expression z^c means e^clnx where ln x is the elementary real natural logarithm (that is, the principal value Ln z (z = x > 0) in the sense of our definition). Also, if z = e, the base of the natural logarithm, z^c = e^c is conventionally regarded as the unique value obtained from (1) in Sec. 13.5.

From (7) we see that for any complex number a,

We have now introduced the complex functions needed in practical work, some of them (e^z, cos z, sin z, cosh z, sinh z) entire (Sec. 13.5), some of them (tan z, cot z, tanh z, coth z) analytic except at certain points, and one of them (ln z) splitting up into infinitely many functions, each analytic except at 0 and on the negative real axis.

For the inverse trigonometric and hyperbolic functions see the problem set.

CHAPTER 13 REVIEW QUESTIONS AND PROBLEMS

1. Divide 15 + 23i by −3 + 7i. Check the result by multiplication.
2. What happens to a quotient if you take the complex conjugates of the two numbers? If you take the absolute values of the numbers?
3. Write the two numbers in Prob. 1 in polar form. Find the principal values of their arguments.
4. State the definition of the derivative from memory. Explain the big difference from that in calculus.
5. What is an analytic function of a complex variable?
6. Can a function be differentiable at a point without being analytic there? If yes, give an example.
7. State the Cauchy–Riemann equations. Why are they of basic importance?
8. Discuss how e^z, cos z, sin z, cosh z, sinh z are related.
9. ln z is more complicated than ln x. Explain. Give examples.
10. How are general powers defined? Give an example. Convert it to the form x + iy.

11–16 Complex Numbers. Find, in the form x + iy, showing details,

• 11. (2 + 3i)²
• 12. (1 −i)¹⁰
• 13. 1/(4 + 3i)
• 14.
• 15. (1 + i)/(1 − i)
• 16. e^πi/2, e^−πi/2

17–20 Polar Form. Represent in polar form, with the principal argument.

• 17. −4 − 4i
• 18. 12 + i, 12 − i
• 19. −15i
• 20. 0.6 + 0.8i

21–24 Roots. Find and graph all values of:

• 21.
• 22.
• 23.
• 24.

25–30 Analytic Functions. Find f(z) = u(x, y) + iv(x, y) with u or v as given. Check by the Cauchy–Riemann equations for analyticity.

• 25 u = xy
• 26. v = y/(x² + y²)
• 27. v = −e^−2x sin 2y
• 28. u = cos 3x cosh 3y
• 29. u = exp(−(x² − y²)/2) cos xy
• 30. v = cos 2x sinh 2y

31–35 Special Function Values. Find the value of:

• 31. cos (3 − i)
• 32. Ln (0.6 + 0.8i)
• 33. tan i
• 34. sinh (1 + πi), sin(1 + πi)
• 35. cosh (π + πi)

SUMMARY OF CHAPTER 13 Complex Numbers and Functions. Complex Differentiation