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AMAT 415: Basic Complex Analysis

Michael P. Lamoureux

February 4, 2008

1 Complex numbers

Complex numbers are numbers of the form x + yi, where x and y are real numbers, and i is the square root of minus one. For instance, 2 + 3i is a complex number, as is

√ 2 + πi. We can add,

subtract, multiply, and divide complex numbers just like regular real numbers, with the usual laws of algebra holding (commutative, associative, distributive laws, etc). We usually write z = x + yi for a generic complex number, where x is the real part of the number, and y is the imaginary part of the number. It is convenient to define the basic functions

Division by a complex number is only slightly more complicated, just use the formula

z1 z2

= z1z2 |z2|2

(10)

so, for instance 2 + 3i 4 + 5i

= (2 + 3i)(4− 5i)

|4 + 5i|2 =

23 + 2i 16 + 25

= 23 41

+ 2 41

i. (11)

If this seems weird to you, remember that the point of division is to be the reverse of multiplication. So we can check the answer, by multiplying(

23 41

+ 2 41

i

) (4 + 5i) = (4∗23−5∗2)/41+((23∗5+2∗4)/41)i = (82/41)+(123/41)i = 2+3i (12)

which is the original numerator.

The point of all this is that with these definitions, the set of all complex numbers forms a field, which means we can do all the usual arithmetic tricks with this set of numbers. The only thing we don’t have is an order, so it doesn’t make sense to ask whether 2 + 3i < 4 + 5i.

You should also note how the arithmetic operations combine with conjugation. For instance, it is easy to verify that

z1 + z2 = z1 + z2 (13) z1 − z2 = z1 − z2 (14) (z1z2) = z1 z2 (15)

(z1/z2) = z1/z2 (16) z = z (17) z = z if and only if z is real (18)

Figure 1: Two complex numbers on the complex plane.

where the radial distance r = |z| is just the absolute value of z, and φ = arg(z) is the angle measured from the x axis to the radial vector pointing to z. The sum cos(θ) + i sin(θ) comes up so often that we give is a special notation, using the exponential form

eiθ = cos(θ) + i sin(θ). (23)

Note that the usual property of exponentials (exp of a sum is the product of the exp’s) summarizes the familiar trig formulas for the sum of angles. That is, we have three equivalent formulas:

ei(θ1+θ2) = eiθ1eiθ2 (24) cos(θ1 + θ2) + i sin(θ1 + θ2) = [cos(θ1) + i sin(θ1)][cos(θ2) + i sin(θ2)] (25) cos(θ1 + θ2) + i sin(θ1 + θ2) = [cos(θ1) cos(θ2)− sin(θ1) sin(θ2)] + (26)

i[sin(θ1) cos(θ2) + cos(θ1) sin(θ2)] (27)

What is amazing is now you don’t have to remember those complicated sine, cosine laws that you learned in trig: they can all be derived from these simple exponential rules.

The form z = reiθ is called the polar representation of the complex number. Notice we can do multiplication in this form, so

z1z2 = (r1eiθ1)(r2eiθ2) = (r1r2)(eiθ1eiθ2) = (r1r2)ei(θ1+θ2). (28)

Thus we see that when two complex numbers are multiplied, their lengths (abs. value) multiply, and their angles (arg) just add. We can sum up several related properties as follows:

|z1 ∗ z2| = |z1| ∗ |z2| (29) |z1/z2| = |z1|/|z2| (30)

arg(z1 ∗ z2) = arg(z1) + arg(z2) mod 2π (31) arg(z1/z2) = arg(z1)− arg(z2) mod 2π (32)

arg(z) = −arg(z) mod 2π (33)

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These comments “mod 2π” point out one difficulty that the angle from the x axis is not uniquely defined, since we can wrap around the origin many times as we work out the angle. There are several conventions used when defining the argument function. One standard convention is to choose the angle measurement so that we always have

0 ≤ arg(z) < 2π; (34)

another convention is to choose −π < arg(z) ≤ π. (35)

Mathematicians usually use the first convention. Computer scientists, physical scientists, and others, often use the second convention. MATLAB uses the second convention. The problems with calculating modulo 2π remains in any convention you choose.

From the geometric picture, it is tempting to define the arg function as

arg(z) = arctan(y/x), (36)

however, this is only valid on the right half of the complex plane, where x > 0. So be careful.

To summarize, the graphical representation of complex numbers on the two dimensional plane tells us a lot about how complex numbers behave under calculations. Addition and subtraction is just like with vectors. Multiplication of two complex numbers just multiplies their vector lengths, and adds the polar angles. You can think about arithmetic operations on complex numbers as simple geometric movements.

1.3 Algebraic results

Since we know how multiplication works, we have a simple formula for computing powers of a complex number. This is called de Moivre’s formula. With z = reiθ, then

zn = rneinθ for any integer n. (37)

From this, we can also compute the n-th roots of any complex number. Again, with z = reiθ, the n-th roots of z are given as

zk = n √

rei(θ/n+2πk/n), k = 0, 1, 2, . . . , n− 1. (38)

The reason this works of course follows from de Moivre’s formula, since we see that

znk = ( n √

r)nein(θ/n+2πk/n) = reiθ+2πk = reiθ = z. (39)

Figure 2 gives a nice geometric interpretation of this root formula, for the case of finding fourth roots. Notice the four roots are uniformly spread around a circle of fixed radius. This happens in general case of finding n-th roots of a complex number.

The numbers zk = ei2πk/n, k = 0, 1, 2, . . . , n − 1 are called the n-th roots of unity, and are the n distinct solutions to the equation

zn = 1. (40)

4

Figure 2: The complex number z = 4 + 7i and its 4-th roots z0, z1, z2, z3. Note the four roots are uniformly spread around a circle of radius r = 4

√ |z|. The polar angle of z0 is one-quarter the polar

angle of z.

As in the previous example, these n complex roots of unity are uniformly spread around the circle of radius one in the complex plane, and include the trivial real root z0 = 1. We will see these roots of unity many times in the course, in particular with the discrete Fourier transform.

A deep result in algebra, which we can prove using complex analysis, is that EVERY polynomial equation of the form

zn + a1zn−1 + a2zn−2 + · · ·+ an−1z + an = 0 (41)

can be solved, finding roots z1, Z2, . . . , zn to the equation. Thus the polynomial can always be factored in the form

zn + a1zn−1 + a2zn−2 + · · ·+ an−1z + an = (z − z1)(z − z2)(z − z2) · · · (z − zn). (42)

This result is called the Fundamental Theorem of Algebra and holds whether the coefficients a1 . . . an are real or complex. The quadratic formula we learned in high school shows how to solve this in the simple case of n = 2. There are also formulas for n = 3, 4, but for higher degree polynomial, the roots may be found using numerical methods.

2 Elementary functions

We are interested in defining functions that take any complex number z and compute a new complex function. For instance,

f(z) = z3 + 23z2 + (2 + 3i)z + (4 + 5i) (43)

5

is a simple example of a polynomial function that maps complex numbers to complex numbers. A rational function is the quotient of two polynomials, such as the function

f(z) = z3 + 23z2 + (2 + 3i)z + (4 + 5i)

z2 + 1 . (44)

This last function is undefined at the points z = ±i, since we can’t divide by zero. But we still consider it a perfectly useful complex valued function, with a domain that include all but two complex numbers.

The complex exponential function is defined on complex number z = x + iy as

ez = exeiy = ex(cos(y) + i sin(y). (45)

From this, we may define the usual trig functions as

cos(z) = eiz + e−iz

2 (46)

sin(z) = eiz − e−iz

2i (47)

tan(z) = sin(z) cos(z)

= 1 i

eiz − e−iz

eiz + e−iz (48)

sec(z) = 1

cos(z) =

2 eiz + e−iz

(49)

csc(z) = 1

sin(z) =

2i eiz − e−iz

(50)

cot(z) = cos(z) sin(z)

= i

1 eiz + e−iz

eiz − e−iz . (51)

The hyperbolic trig function are defined similarly, with

cosh(z) = ez + e−z

2 (52)

sinh(z) = ez − e−z

2 (53)

tanh(z) = sinh(z) cosh(z)

= ez − e−z

ez + e−z (54)

sech(z) = 1

cosh(z) =

2 ez + e−z

(55)

csch(z) = 1

sinh(z) =

2 ez − e−z

(56)

coth(z) = cosh(z) sinh(z)

= ez + e−z

ez − e−z . (57)

What we see is that all the trig and hyperbolic function are defined in terms of the exponential function. So a careful examination of this one function is in order. We will do that in the next section.

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It is worth checking that the trig identities still hold. For instance, we can check that

sin(z1 + z2) = sin(z1) cos(z2) + cos(z1) sin(z2). (58)

Do this as an exercise, using the definition of sine and cosine above, and the exponential laws.

It is also interesting to note that the trig functions are defined almost exactly the same as the hyperbolic functions, except for the careful placement of a factor of i. Thus, we see that on pure imaginary numbers of the form iy, we have the interesting identities

sin(iy) = i ∗ sinh(y) (59) cos(iy) = cosh(y) (60) tan(iy) = i

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