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# Complex Numbers, Complex Functions and Contour Integrals

## 1. Complex Numbers

The prevalence of the complex numbers throughout the scientific world today
belies their long and rocky

history. Much like the negative numbers, complex numbers were originally viewed
with mistrust and skepticism.

In fact, the term “imaginary number” was a derogatory term coined by René
Descartes, who found such numbers

unsettling. Just as with the negative numbers, however, history has shown that
not only are the complex

numbers immensely useful, they can also have physical meaning, and hence are no
less “real” than the real

numbers. Perhaps more importantly, they have given deep insight into a broad
range of problems and become

an indispensable tool both for mathematicians and scientists in general.

Complex numbers first gained prominence in the 16^{th}
century, when Italian mathematicians Niccolo Tartaglia

and Gerolamo Cardano discovered formulas for the roots of general cubic and
quartic polynomials. Surpisingly,

their formulas required the use of square roots of negative numbers, even when
using the formulas to find real

roots of polynomials. You’ve likely encountered similar situations in your own
studies when trying to find roots

of quadratic polynomials. Recall the following familiar theorem:

**Quadratic Formula. **Suppose f(x) = ax^{2} + bx + c
with a, b, c ∈ R and a ≠0. If b^{2} − 4ac < 0, then f has

no real roots. If b^{2} − 4ac ≥ 0, then the roots of f are precisely

The quantity b^{2} − 4ac is sometimes called the discriminant of f, and written
disc(f).

**Example.**

Suppose f(x) = x

^{2}+ 1. Then disc(f) = −4 < 0, so f has no real roots.

This example is not purely artificial. In fact, you might
have come across this scenario when searching for

eigenvalues of linear transformations.

**Example.**

Suppose T : R

^{2}→ R

^{2}is the linear transformation which rotates the plane 90°counterclockwise about the

origin. The matrix for T with respect to the standard basis is

The characteristic polynomial of this matrix is By the previous example, it follows that A

has no real eigenvalues.

If we restrict ourselves to the real numbers, we see that
not every polynomial has a root (and hence not every

linear transformation has an eigenvalue). This seems quite unfortunate,
especially since the quadratic formula

seems to suggest what the roots (or eigenvalues) “should” be. To remedy this
situation, we will enlarge our

collection of numbers to include some of these “imaginary” roots. We’ll start by
simply adding

**Definition.** Let i represent a “number” which satisfies i^{2} = −1. We call
i the imaginary unit.

We’ll now add i to our collection of numbers.

**Definition.** We define the complex numbers to be the
set

C = {a + ib | a, b ∈ R}.

We define addition and multiplication of complex numbers
using the relation i^{2} = −1. That is, given complex

numbers a_{1} + ib_{1}, a_{2} + ib_{2}, we define

(a_{1} + ib_{1}) + (a_{2} + ib_{2}) := (a_{1} + a_{2}) + i(b_{1} + b_{2})

and

(a_{1} + ib_{1})(a_{2} + ib_{2}) := (a_{1}a_{2} − b_{1}b_{2}) + i(a_{1}b_{2} + a_{2}b_{1}).

**Examples.**

(1) Suppose f(x) = x

^{2}+ 4. We claim the roots of f are x = ±2i. Indeed, observe that

f(±2i) = (±2i)

^{2}+ 4 = 4i

^{2}+ 4 = 4(−1) + 4 = 0.

Note that this agrees with the roots predicted by the quadratic formula:

±2i.

(2) Suppose f(x) = x

^{2}+2x+3. Using the quadratic formula, the predicted roots of f are

As a check, we compute

MatLab can natively work with complex numbers. For example, the above computation is entered

as follows:

**>> (-1+sqrt(2)*i)^2+2*(-1+sqrt(2)*i)+3
ans =**

**-4.4409e-016**

Notice that MatLab has introduced a small rounding
error, outputting a result of −4.4409x10^{-16} ≈ 0.

**Exercise 1.** Use MatLab to compute (1 − 2i)^{5}.

You might be worried we will need to keep adding new “numbers” to our collection
as we attempt to find

roots of polynomials of higher degrees. The following theorem, known as the
Fundamental Theorem of Algebra,

ensures we won’t need to extend our collection any further.

**Theorem.** Let p(x) = a_{n}x^{n} + · · · + a_{1}x + a_{0} be a polynomial, with a_{i} ∈ C and a_{n}≠ 0. Then

p(x) = a_{n}(x − r_{1}) · · · (x − r_{n})

for some r_{1}, . . . , r_{n} ∈ C.

In other words, when working with the complex numbers, every polynomial of
degree n has n (not necessarily

distinct) roots. This theorem may seem more amazing in light of the fact there
do not exist formulas for the

roots of general polynomials of degree five or more; that is, we can’t prove the
theorem by simply writing down

the roots of a general polynomial and checking they’re all in our set C.

**Example.**

The roots of x^{2} + 1 are x = ±i, which we can express as

x^{2} + 1 = (x − i)(x + i).

In general, finding the roots of a polynomial is quite difficult. Fortunately,
MatLab can do this for us. First

we must define our polynomial. MatLab stores polynomials as row vectors, with
components given by the

coefficients of the monomials. For example, to store the polynomial p(x) = x^{5} −
2x + 3, we enter

**>> p=[1 0 0 0 -2 3];**

To find the roots of p we use the roots command:

**>> r=roots(p)**

**r =
-1.4236
-0.2467 + 1.3208i
-0.2467 - 1.3208i
0.9585 + 0.4984i
0.9585 - 0.4984i**

As predicted by the previous theorem, the degree five polynomial p has five complex roots.

**Exercise 2. **Use MatLab to find the roots of the polynomial p(x) = x^{4} − 2x^{3} + 1.

Since we’ll be working with the complex numbers, it will be useful to have a few
additional definitions.

**Definition.** Suppose z ∈ C is given by z = a + ib, with a, b ∈ R. We make the
following definitions:

(1) The real number a is called the real part of z, and is denoted Re(z). A
complex number z with Re(z) = 0

is called purely imaginary.

(2) The real number b is called the imaginary part of z, and is denoted Im(z). A
complex number z with

Im(z) = 0 is called purely real.

(3) The complex conjugate of z, denoted by
is given by

= a − ib = Re(z) − iIm(z).

**Example.**MatLab has a built-in function for conjugation, called conj. For example:

**>> conj(-1+2*i)
ans =
-1.0000 - 2.0000i**

**Exercise 3.** Prove that Re(z) =
and Im(z) =

By the above exercise, any equation involving the real and imaginary parts of z
can be written in terms of z

and This observation will be useful to remember when we discuss complex
functions in Section 2.

It is often helpful to picture the complex numbers as lying in the plane, with
the real part along the horizontal

axis and the imaginary part along the vertical axis.

Unsurprisingly, this plane is called the complex plane. It was first described
by Caspar Wessel in 1799,

although it is also sometimes referred to as the Argand plane and credited to
Jean-Robert Argand. Use of

the complex plane was later popularized by Carl Gauss, and it was only after
this geometric interpretation

was introduced that the complex numbers became widely accepted. The complex
plane can be used to give

polar coordinates for z, with which one can deduce many beautiful relations.
Unfortunately, this would take us

beyond the scope of the current assignment.