Download 1. Complex Numbers and the Complex Exponential

1. Complex Numbers and the Complex Exponential
It would be possible to express some of the basic concepts of Fourier analysis entirely in real
terms, with sines and cosines and real coefficients. Possible, but awkward and severely limited. For
reasons that will become increasingly clear as the course progresses, we will use the language of
complex numbers and we will insist that the students become both comfortable and familiar with
this language.
Please note that we are not asking for the knowledge contained in a complex analysis course
(such as MATH 461). We will be using complex-valued functions of real or integer variables, whereas
the best and strongest material of the complex analysis course concerns complex-valued functions
of a complex variable. The material were interested in is more basic, and could be presented at the
level of Calculus II.
1.1.
The complex numbers
The set of all complex numbers (conventionally denoted by a fancy letter C: C, or C, or C)
is the set of everything that can be written in the form x + yi, where x and y are real numbers.
Addition and multiplication are performed as they would be for polynomials in the character “i”
with the understanding that i2 = −1. This could all be done on a more formal level by defining an
addition and a multiplication on ordered pairs of real numbers and verifying that these operations
satisfy the appropriate algebraic laws (which are those of a field). Although we will not carry out
that plan, we will adopt the geometric consequences: since each complex number is a pair of real
numbers, we will plot complex numbers on a plane, just as we would plot vectors in R2 .
Both the addition and the multiplication of complex numbers have geometric interpretations.
In the case of addition, this interpretation is exactly the same (the parallelogram picture) as for
vector addition in R2 . That is, (a + bi) + (c + di) = (a + c) + (b + d)i. It is the multiplication that
distinguishes C from R2 . We’ll study the geometric meaning of multiplication later. For now, note
that (a + bi)(c + di) = (ac − bd) + (ad + bc)i.
1.2.
Real parts, imaginary parts, and complex conjugates
Suppose a complex number z is written in its “standard form”: z = x + yi, where x and y are
real. Then x is the real part of z, written as Re z and y is the imaginary part of z, written as Im z.
Please note that both Re z and Im z, especially Im z, are real numbers. The complex conjugate of
z is written z, and is defined as z = x − yi. The following is a list of identities involving these
concepts; I will leave it to the reader to verify them.
z = Re z − i Im z
(1.1a)
z =z
(1.1b)
Re(z + w) = Re z + Re w and Im(z + w) = Im z + Im w
(1.1c)
Re zw = (Re z)(Re w) − (Im z)(Im w)
(1.1d)
1
Im zw = (Re z)(Im w) + (Im z)(Re w)
(1.1e)
Re z =
z+z
2
(1.1f)
Im z =
z−z
2i
(1.1g)
(z + w) = z + w
(1.1h)
zw = z w
(1.1i)
A complex number z such that Im z = 0 is a number that can be written as z = x + 0i = x.
The set of all such numbers lies in a one-to-one correspondence with the real numbers, and the
arithmetic of such numbers is indistinguishable from the arithmetic of the real numbers. We go
ahead and call such a complex number real. On the complex plane, the set of all such numbers
(the complex numbers which are real) comprises the horizontal axis, also known as the real axis.
A complex number z such that Re z = 0 is a number that can be written as z = 0+yi = yi. Such
a complex number is called a purely imaginary number, and set of all such numbers geometrically
comprises the vertical axis, also known as the imaginary axis.
The only complex number which is both real and purely imaginary is 0. On the other hand, most
complex numbers do not lie on either axis. Although the official meaning of the words “complex
number” is any member of the set C, these words are sometimes informally used to mean a member
of C which is neither real nor purely imaginary.
1.3.
Absolute values and inequalities
What happens if you multiply a complex number and its conjugate? If z = x + iy, then
zz = (x + iy)(x − iy) = x2 − (iy)2 = x2 − i2 y 2 = x2 + y 2
(1.2)
This is a nonnegative real number (positive unless z = 0) which we recognize from Euclidean
analytic geometry as the square of the distance from the origin to the point (x, y) (i.e., the point
in R2 that we identify with the complex number z = x + iy). This insight, together with the notion
from the real numbers that absolute value represents distance, leads us to define the absolute value
or modulus of a complex number.
Definition:
|z| =
√
zz =
p
p
x2 + y 2 = (Re z)2 + (Im z)2
(1.3)
This calculation gives us one of the most basic tricks of complex arithmetic: if we have an
quotient with a complex denominator which we want to express in standard form, we multiply the
numerator and denominator of this fraction by the complex conjugate of the denominator.
Example:
Find the imaginary part of
3 − 4i
:
−1 + 2i
3 − 4i
(3 − 4i)(−1 − 2i)
−3 + 4i − 6i + 8i2
−11 − 2i
=
=
=
2
2
−1 + 2i
(−1 + 2i)(−1 − 2i)
(−1) + 2
5
2
2
Hence, the required imaginary part is − .
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Since z = w if and only if Re z = Re w and Im z = Im w, one complex equation contains as
much information as two real equations. The same can’t be said for inequalities. In fact, (and
this is very important), complex numbers don’t have inequalities. Nevertheless, inequalities are
quite important in working with complex numbers, with the proviso that only real numbers have
inequalities. The complex number z may not itself be real, but Re z, Im z, and |z| are all real and
may all be included in inequalities. In fact, if our claim in a theorem is an inequality for which
one side is some not-obviously-real arithmetic combination of complex numbers, then part of the
conclusion of the theorem is that that side of the inequality is real after all.
Let’s list some basic facts about absolute values and inequalities, together with an item of old
business:
z
|z|
|zw| = |z| |w| and =
w
|w|
z
z
=
w
w
(1.4a)
(1.4b)
|z| = |z|
(1.4c)
−|z| ≤ Re z ≤ |z| and − |z| ≤ Im z ≤ |z|
(1.4d)
|z + w| ≤ |z| + |w|
(1.4e)
We leave most of the derivations of these identities to the reader; however, (1.4e) (also known as
the triangle inequality) merits special attention. In order to prove it, we start with a side inequality,
using (1.1f), (1.4a), (1.4c) and (1.4d) along the way.
zw + zw = zw + zw = 2 Re zw ≤ |zw| = 2|z| |w|
(1.5)
Inequality (1.5) now appears in the middle of the proof of the triangle inequality.
|z + w|2 = (z + w)(z + w) = zz + zw + zw + ww
= |z|2 + zw + zw + |w|2 ≤ |z|2 + 2|z| |w| + |w|2 = (|z| + |w|)2
(1.6)
Talking square roots in (1.6) leaves us with our result.
1.4.
The complex exponential
We can write infinite series (sums) with complex terms. In doing so, everything that we learned
about series with real terms in Calculus II remains true. In particular, if a series converges absolutely, then it converges. Furthermore, absolute convergence can often be determined by a variety
of comparison-like tests, including direct or limit comparison, the integral test, the ratio test, or
the root test. (However, you will see more conditionally convergent series in this course than you
have ever seen before.) One can write power series in z. Each such power series has a radius of
convergence, which is a name that makes better sense in the complex numbers than it ever did in
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the real numbers. Were not going to pursue complex power series in general or the question of just
what functions have power series (those matters lie near the heart of a complex analysis course
such as MATH 461), but we are interested in one very special series.
Definition:
z
e =
∞
X
zn
n=0
n!
=1+z+
z2 z3 z4 z5
+
+
+
+ ...
2
3!
4!
5!
(1.7)
A quick application of the ratio test shows us that this series converges for all z ∈ C. If z is
real, this is exactly the exponential that we are familiar with. In general, it is possible to show
that for all complex z and w, ez+w = ez ew . This means that to understand ez in general, we must
understand ex+iy = ex eiy . Since we already understand ex , this leaves us with eiy to deal with.
Let’s expand out the power series to see what we get.
eiy = 1 + iy +
(iy)2 (iy)3 (iy)4 (iy)5 (iy)6 (iy)7
+
+
+
+
+
+ ···
2
3!
4!
5!
6!
7!
y 2 iy 3 y 4 iy 5 y 6 iy 7
−
+
+
−
−
+ ···
2
3!
4!
5!
6!
7!
y3 y5 y7
y2 y4 y6
+
−
··· + i y −
+
−
+ ···
= 1−
2
4!
6!
3!
5!
7!
= 1 + iy −
Recognizing each of these power series for what they are, we write Euler’s Identity:
eiy = cos y + i sin y
(1.8)
Thus the sine, the cosine, and the exponential that we knew from real-variable calculus stand
revealed as aspects of a single function. The following identities (true for all real x) may be useful.
cos x =
eix + e−ix
= Re(eix )
2
(1.9)
sin x =
eix − e−ix
= Im(eix )
2i
(1.10)
|eix | = 1
1.5.
(1.11)
Polar coordinates, DeMoivre’s Theorem, and roots of unity
We geometrically identify the complex number z = x + iy with the point (x, y) ∈ R2 . To use
x and y (that is, Re z and Im z) to represent this point is to use rectangular coordinates. But
we know that this point can also be written in terms of the polar coordinates r and θ. One set of
formulas for converting between rectangular and polar coordinates is the following:
(
x = r cos θ
(1.12)
y = r sin θ
But then z = z + iy = r cos θ + ir sin θ = r(cos θ + i sin θ) = reiθ . If we recallphow to reverse the
equations (1.12), we find that the r is uniquely determined by x and y: r = x2 + y 2 . We have
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met this quantity before, as the modulus or absolute value of z: r = |z|. On the other hand, θ is not
uniquely determined in any particularly natural way. We can force it to be uniquely determined
by convention, but even then, we won’t always agree on the appropriate convention. Sometimes
we want θ ∈ [0, 2π), sometimes we want θ ∈ [−π, π), and sometimes we want something else. The
following is a loose, sloppy definition, but it will do for a while.
“Definition”:
The argument of z, written arg z, is that θ such that |z| = reiθ .
Now suppose that z = reiα and w = seiβ . Then
zw = (reiα )(seiβ ) = rseiα eiβ = rsei(α+β)
(1.13)
We note from this that |zw| = |z| |w|, which we already knew, and
arg(zw) = arg z + arg w
(1.14)
We are now in position to give the geometric interpretation of complex multiplication: when you
multiply two complex numbers, you multiply the absolute values and add the arguments. Complex
addition may be most simply expressed in rectangular coordinates; complex multiplication may be
most simply expressed in polar coordinates.
An inductive extension of a special case of (25) has a special name:
DeMoivre’s Theorem:
If z = reiθ , then z n = rn einθ
(1.15)
DeMoivre’s Theorem is the key to finding the nth roots of a complex number w : that is, the
solutions of the equation z n = w. There is a very general theorem, known as the Fundamental
Theorem of Algebra that assures us that every complex polynomial can be completely factored into
linear factors, or, equivalently, every nth degree polynomial has (counting multiplicities) exactly
n complex roots. That theorem does not belong in this course; it belongs in a complex analysis
course such as Math 461 or 562A. We will not concern ourselves with the general theorem, but we
will be specifically interested in the n distinct solutions of z n = w. One way to phrase this is to say
that every nonzero complex number has two square roots, three cube roots, four 4th roots and so
on. We are most interested in the nth roots of 1, also known as the roots of unity.
Claim:
Let N be a positive integer. Then the following N numbers
1 (= e2πi·0 ), e2πi/N , e2πi·2/N , . . . , e2πij/N , . . . , e2πi(N −1)/N
(1.16)
are the N distinct solutions to the equation z N = 1.
Furthermore, these N numbers all lie on the unit circle {z : |z| = 1} and (for N ≥ 3) form the
vertices of a regular polygon with N sides.
In a few special cases the N th roots of unity can be expressed using the values of the trigonometric functions of the special angles emphasized in trigonometry and calculus courses. This
is completely true only
the six 6th roots of 1 √are
√ for N = 1, 2, 3,√4, 6, 8, or 12. For example, √
1 i 3 2πi/3
1 i 3 πi
1 i 3
1 i 3
e0 = 1, eπi/3 = +
,e
=− +
, e = −1, e4πi/3 = − −
, and e5πi/3 = −
.
2
2
2
2
2
2
2
2 ◦
On the other hand, if N = 5, we must evaluate the trigonometric functions of such angles as 72
and 144◦ to express these roots. (It is possible to evaluate those particular values – the trigonometric functions of the pentagon angles – exactly using square roots, but they’re not emphasized
or memorized in trigonometry courses.)
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Fix N for the time being. Let ζ be the first non-trivial root on the list in (1.16). That is, let
ζ = e2πi/N
(1.17)
Then the list of all of the N th roots of one is simply {ζ j : 0 ≤ j < N }. That is, all of the roots can
be realized as powers of this first root.
What are the solutions of the equation
z N = w for w 6= 1? If we write w in polar coordinate
√
as w = seiα , then one solution is z0 = N s eiα/N . We can then find all of the solutions by letting
z = zj = z0 ζ j for 0 ≤ j < N, where ζ is as in (1.17). As an example of this, lets find the square
π
roots of i. In polar coordinates, i = 1 · eπi/2 . Thus s = 1 and its square root is one, while α = ,
2
α
π
1
i
so that = . Hence z0 = √ + √ . We now multiply this by the two square roots of 1, namely
2
4
2
2
1
i
1
i
1 and −1, to get the two square roots of i : √ + √ and − √ − √ .
2
2
2
2
1.6.
Some calculus uses
You should all have taken a differential equations course (Math 364A or 370A) and in that course
you should have made a great deal of use of the complex exponential. As one further illustration,
let’s work a calculus problem.Z
∞
Example:
Evaluate
e−ax cos bx dx for a > 0.
0
This problem is done earlier in Calculus II by integrating by parts twice and solving for the
original integral. We do it now by noting (by (21 that cos bx = Re(eibx ) so that the integrand is
the real part of e(−a+bi)x . Furthermore, the integral of a real part is the real part of an integral. (If
you believe (3), you should believe this.) Hence,
Z ∞
Z ∞
e−ax cos bx dx = Re
e(−a+bi)x dx
0
0
∞ 1
1
(−a+bi)x = Re
e
= Re 0 −
−a + bi
−a + bi
0
1
a + bi
a + bi
a
= Re
·
= Re 2
= 2
.
a − bi a + bi
a + b2
a + b2
For no more work than this, we also get that
Z ∞
e−ax sin bx dx =
0
1.7.
Exercises
Derivations:
1.1 Derive the identities (1.1f), (1.1g), and (1.1i).
1.2 Derive the identities (1.9), (1,10), and (1.11).
Calculations:
6
a2
b
.
+ b2
1.3. Evaluate the following expressions and write your answers in the form a + bi.
2 + 5i
a) (2 + 5i)(4 − i)
b)
c) i100
d) 2i(3 − i)
4−i
1
1.4. Find Im
(2 + 3i)2
1.5. Find the fourth degree polynomial with roots 1 + i, 1 − i, −1 + i, and −1 − i.
1.6. Find all complex solutions to z 3 + 8 = 0.
1.7. Let f (t) = (2 + 3i)e(−4+5i)t
a) Find Re(f (t)) and Im(f (t))
1.8. Let N be a positive integer. Compute
b) Find |f (t)|
N
−1
X
e2πik/N . (Hint: finite geometric sum.)
k=0
Z
2π
e3ix dx
1.9. Find
0
Z
1.10. Find
∞
xe−3x cos 2x dx (Hint: write this as the real part of something.)
0
Supplementary exercises:
1.11. (Makes use of ideas and material from Math 361A.)
(a) If (zn ) is a sequence in C and z ∈ C, give the formal definition of lim zn = z.
n→∞
(b) Prove that lim zn = z if and only if lim Re zn = Re z and lim Im zn = Im z.
n→∞
n→∞
n→∞
(c) Define a Cauchy sequence in C, and prove that a sequence in C converges if and only if
it is Cauchy.
(d) Prove that an absolutely convergent series of complex numbers converges.
1.12. Expand out the power series to show that for all z, w ∈ C, ez+w = ez ew . Justify your steps.
(Hint: you’ll need the binomial theorem.)
1.13. (Some background in algebra, e.g., MATH 444, helps clarify the purpose of this problem.)
Let M2 (R) be the set (ring) of all 2 × 2 matrices with real entries, with the usual definitions
of matrix addition and matrix multiplication. Define φ : C → M2 (R) by
x −y
φ(z) = φ(x + yi) =
.
y x
(a) Show that for all z, w ∈ C, φ(z + w) = φ(z) + φ(w) and φ(zw) = φ(z)φ(w).
(b) Show that φ is one-to-one.
(c) Show that for all z 6= 0, the matrix φ(z) is invertible. Calculate its inverse.
0 −1
(d) Find two or three interesting things to say about the matrix
. In particular,
1 0
note its eigenvalues, note its multiplicative properties, note how it is related to φ.
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[Context: this shows that we can find an isomorphic copy of the field of complex numbers as
a subring within M2 (R).]
1.14. This exercise discovers something about compass-and-straightedge constructions. The “constructible numbers” are the numbers that can be obtained from the integers by taking
a finite
2π
number of sums, differences, products, quotients, and square roots. Let x = cos
=
5
2π
cos(72◦ ), let y = sin
= sin(72◦ ), and let z = x + iy = e2πi/5 . We want to show that x
5
and y are constructible and hence that a regular pentagon may be constructed with compass
and straightedge. We will do this by computing exact values for x and y.
(a) Show that z satisfies the equation 1 + z + z 2 + z 3 + z 4 = 0.
(b) In the equation in (a), group z and z 4 together and group z 2 and z 3 together. Note that
these are complex conjugate pairs. Use this grouping to obtain a real quadratic equation
for x and y, and substitute for y so as to obtain a quadratic equation for x.
(c) Find exact values for x and y.
(d) Use a calculator to show that these exact values agree (to within calculator accuracy)
with the numbers computed directly using the trigonometric functions.
(e) Calculate the exact values of the perimeter and the area of a regular pentagon inscribed
in a circle of radius 1.
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