Download 3. Complex Numbers

MH2801: Complex Methods for the Sciences
Y. D. Chong (2016)
3. Complex Numbers
The imaginary unit, denoted i, is a hypothetical solution to the quadratic equation
z 2 + 1 = 0,
(1)
√
which is an equation that lacks real solutions. In other words, i = −1.
We can let the imaginary unit take part in the usual arithmetic operations of addition
and multiplication, treating it as an algebraic quantity on the same footing as the more
familiar real numbers. Thus, we deal with numbers containing both real and imaginary parts,
called complex numbers. It is one of the most profound discoveries of mathematics that
this seemingly arbitrary idea gives rise to powerful computational methods for addressing
mathematical and physical problems.
3.1
Complex algebra
For any complex number z, we can write
z = x + iy,
(2)
where x and y are real numbers that depend uniquely on z. We refer to these as the real
part of z and the imaginary part of z, respectively. The real and imaginary parts are also
commonly denoted as Re(z) and Im(z), respectively, where the Re and Im operations can
be regarded as functions mapping complex numbers to real numbers
The set of complex numbers is denoted by C. We can define algebraic operations on
complex numbers—addition/subtraction, products, and taking powers—simply by following
the usual rules of algebra and setting i2 = −1 whenever it shows up.
Example
Let z = x + iy, where x, y ∈ R. What is z 2 ?
z 2 = (x + iy)2
2
= x + 2x(iy) + (iy)
2
(3)
2
2
= x − y + 2ixy
(4)
(5)
Therefore, the real and imaginary parts are:
Re(z 2 ) = x2 − y 2 , Im(z 2 ) = 2xy.
(6)
There’s one caveat: for now, we’ll only consider taking integer powers, such as z −1 or
z 2 . Taking non-integer powers, such as z 1/3 , introduces vexatious complications which we’ll
postpone for now (this will be dealt with when discussing branch points and branch cuts).
Another interesting fact, which can be easily proven, is that real coefficients can be freely
moved into or out of Re(· · · ) and Im(· · · ) operations:
Re(αz + βz 0 ) = α Re(z) + β Re(z 0 )
for α, β ∈ R.
(7)
Im(αz + βz 0 ) = α Im(z) + β Im(z 0 )
This has an important consequence: if we have a complex function of a real variable, then we
can calculate the derivative of that function from the derivatives of the real and imaginary
parts. This is shown in the example below.
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MH2801: Complex Methods for the Sciences
Y. D. Chong (2016)
Example
If z(t) is a complex function of a real input t, then
dz
d
dz
d
Re
= Re [z(t)] , and Im
= Im [z(t)] .
dt
dt
dt
dt
(8)
This can be proven using the definition of the derivative:
dz
z(t + δt) − z(t)
Re
= Re lim
δt→0
dt
δt
Re[z(t + δt)] − Re[z(t)]
= lim
δt→0
δt
d
= Re [z(t)] .
dt
The Im case works out similarly. Note that the infinitesimal quantity δt is real; otherwise, this wouldn’t work.
3.2
Conjugates and Magnitudes
For each complex number z = x + iy, we define its complex conjugate as a complex
number whose imaginary part has its sign flipped:
z ∗ = x − iy.
(9)
We can show that conjugation obeys two important properties:
(z1 + z2 )∗ = z1∗ + z2∗
(10)
(z1 z2 )∗ = z1∗ z2∗ .
(11)
Example
Let us prove that (z1 z2 )∗ = z1∗ z2∗ . First, let z1 = x1 + iy1 and z2 = x2 + iy2 . Then,
∗
(z1 z2 )∗ = [(x1 + iy1 )(x2 + iy2 )]
∗
= [(x1 x2 − y1 y2 ) + i (x1 y2 + y1 x2 )]
= (x1 x2 − y1 y2 ) − i (x1 y2 + y1 x2 )
= (x1 − iy1 ) (x2 − iy2 )
= z1∗ z2∗ .
For a complex number z = x + iy, we define the magnitude of z as
p
|z| = x2 + y 2 .
(12)
This is a non-negative real number. A complex number and its conjugate have the same
magnitude: |z| = |z ∗ |. Also, we can show that magnitudes have the property
|z1 z2 | = |z1 | |z2 |.
(13)
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MH2801: Complex Methods for the Sciences
Y. D. Chong (2016)
This property is similar to the “absolute value” operation for real numbers, hence the similar
notation. As a corollary, we can show that taking a power of a complex number raises its
magnitude by the same power:
|z n | = |z|n for n ∈ Z.
3.3
(14)
Euler’s formula
Euler’s formula is an extremely important result which states that
eiz = cos(z) + i sin(z).
(15)
This can be proven using the series definition of the exponential function, which is
exp(z) = 1 + z +
z3
z4
z5
z6
z2
+
+
+
+
+ ···
2!
3!
4!
5!
6!
(16)
Previously, we assumed that the input to the exponential function was a real number. However, since complex numbers can be added and multiplied using the same rules of algebra
as real numbers, we can employ this series formula as the definition of the complex exponential function. This is a function that takes complex inputs and gives complex outputs.
When the input happens to be real, the complex exponential function gives the same output
as the real exponential function.
Plugging iz as the input to the complex exponential function gives
(iz)3
(iz)4
(iz)5
(iz)6
(iz)2
+
+
+
+
+ ···
2!
3!
4!
5!
6!
z2
z3
z4
z5
z6
= 1 + iz −
−i +
+i −
+ ···
2!
3!
4! 5! 6!
z2
z4
z6
z3
z5
z7
= 1−
+
−
+ ··· + i z −
+
−
+ ··· .
2!
4!
6!
3!
5!
7!
exp(iz) = 1 + (iz) +
(17)
(18)
(19)
Now, compare the two terms in parentheses to the series expansions for the cosine and sine
functions. We can define the complex cosine and complex sine functions using these
complex series formulas:
z2
z4
z6
+
−
+ ···
2!
4!
6!
z3
z5
z7
sin(z) = z −
+
−
+ ···
3!
5!
7!
cos(z) = 1 −
(20)
(21)
These are perfect matches for the two terms in the series expansion of the complex exponential! Hence, Euler’s formula immediately follows:
eiz = cos(z) + i sin(z).
One important consequence of Euler’s formula is that
iθ q
e = cos2 (θ) + sin2 (θ) = 1
(22)
for θ ∈ R.
(23)
Another interesting consequence is that
eiπ = −1,
(24)
which is a formula that relates two transcendental constants e = 2.7182818285 . . . and
π = 3.141592654 . . . , by means of the imaginary unit. (We saw a different relationship
between these two constants when solving the Gaussian integral.)
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MH2801: Complex Methods for the Sciences
Y. D. Chong (2016)
3.4
The complex plane
A convenient device for conceptualizing complex numbers is to think of a complex number
as a point on a two-dimensional plane, as shown below. This is called the complex plane.
The real and imaginary parts are represented by horizontal and vertical Cartesian coordinates. The coordinate axes are called the “real axis” and the “imaginary axis”, respectively.
3.4.1
Polar representation
If we think of a complex number as a point on the complex plane, its position can also be
represented using polar coordinates instead of Cartesian coordinates. For a complex number
z = x + iy, we can introduce polar coordinates r and θ (both real numbers), such that
p
(25)
r = x2 + y 2 , θ = tan−1 (y/x).
Conversely,
x = r cos θ,
y = r sin θ.
(26)
These are the usual formulas for performing a change of coordinate between two-dimensional
Cartesian coordinates and polar coordinates. Their graphical meaning is shown below:
The radial coordinate is r, and by its definition we see that it is equal to what we have
defined as the magnitude of the complex number: r = |z|. The azimuthal coordinate θ is
called the argument of the complex number, which we sometimes denote by arg(z).
Using Euler’s formula, we can write
z = r cos(θ) + ir sin(θ)
(27)
= r [cos(θ) + i sin(θ)]
(28)
iθ
= re .
(29)
Thus, whenever we can manipulate a complex number into a form AeiB , where A and B are
real, then A is the magnitude and B is the argument. This is used in the following example:
Example
For z ∈ C, it can be shown that the magnitude and argument of exp(z) are:
|exp(z)| = eRe(z) ,
arg [exp(z)] = Im(z).
(30)
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MH2801: Complex Methods for the Sciences
Y. D. Chong (2016)
Proof: Let z = x + iy, where x, y ∈ R; then
ez = ex+iy = ex eiy .
(31)
By inspection, the magnitude of this complex number is ex , and its argument is y.
3.4.2
Geometrical interpretation of complex operations
The complex plane provides useful geometric interpretations for complex algebra operations:
• Addition of two complex numbers can be interpreted as the addition of two coordinate
vectors. If z1 = x1 + iy1 and z2 = x2 + iy2 , then z1 + z2 = (x1 + x2 ) + i (y1 + y2 ) .
Hence, the point corresponding to z1 + z2 is obtained by adding the two coordinate
vectors corresponding to z1 and z2 . From this, we can geometrically prove a useful
inequality relation between complex numbers, called the “triangle inequality”:
|z1 + z2 | ≤ |z1 | + |z2 |.
(32)
• Complex multiplication can be interpreted as a scaling together with a rotation. If
z1 = r1 eiθ1 and z2 = r2 eiθ2 , then z1 z2 = (r1 r2 ) exp[i(θ1 +θ2 )]. The point corresponding
to z1 z2 is obtained by scaling the z1 coordinate vector by a factor of |z2 |, and rotating
it by an angle of θ2 around the origin. In particular, multiplication by eiθ is equivalent
to a pure rotation of angle θ.
• Complex conjugation is equivalent to reflection about the real axis. This moves a point
from the “upper half” of the complex plane to the “lower half”, or vice versa.
3.5
Complex functions
When deriving Euler’s formula, we introduced complex functions defined by taking real
mathematical functions, such as the exponential function, and making them accept complex
number inputs. Let us take a closer look at how these complex functions behave.
3.5.1
Complex trigonometric functions
When we derived Euler’s formula, we noted that it is valid for arbitrary real numbers:
exp(iz) = cos(z) + i sin(z),
for z ∈ C.
(33)
The cosine and sine functions on the right-hand side of this equation are complex trigonometric functions, defined using the same series expansions as the real cosine and sine functions,
except that the inputs z are allowed to be complex numbers:
z3
z5
z7
+
−
+ ···
3!
5!
7!
z2
z4
z6
cos(z) = 1 −
+
−
+ ··· ,
2!
4!
6!
sin(z) = z −
(34)
z∈C
(35)
Note that the outputs of the complex trigonometric functions are complex numbers too.
Thus, some of the familiar properties of the real trigonometric functions don’t apply. For
instance, | sin(z)| and | cos(z)| are not bounded by 1 when z is not real. For example,
(36)
sin(i) = 1.1752 . . . > 1.
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MH2801: Complex Methods for the Sciences
Y. D. Chong (2016)
We can also write the complex cosine and sine functions in terms of the exponential:
1 iz
e + e−iz
2
i
sin(z) = − eiz − e−iz .
2
cos(z) =
(37)
(38)
This is often a convenient step when solving integrals, as shown in the following example.
Example
Consider the (real) integral
Z
∞
I=
dx e−x cos(x).
(39)
0
One way to solve this is to use integration by parts, but another way is to use the
complex expansion of the cosine function:
Z ∞
1 ix
I=
dx e−x
(40)
e + e−ix
2
0
Z ∞
h
i
1
=
dx e(−1+i)x + e(−1−i)x
(41)
2 0
∞
e(−1−i)x
1 e(−1+i)x
(42)
+
=
2 −1 + i
−1 − i 0
1
1
1
=−
+
(43)
2 −1 + i −1 − i
1
= .
(44)
2
3.5.2
Complex trigonometric identities
Euler’s formula provides a convenient way to deal with trigonometric functions. Consider
the addition formulas
sin(z1 + z2 ) = sin(z1 ) cos(z2 ) + cos(z1 ) sin(z2 )
(45)
cos(z1 + z2 ) = cos(z1 ) cos(z2 ) − sin(z1 ) sin(z2 ).
(46)
The standard proofs for these formulas are geometric: you draw a figure, and solve a bunch of
relations between the angles and sides of the various triangles, making use of the Pythagorean
formula. But using the Euler formula, we can prove these algebraically. For example,
1 iz1
e + e−iz1 eiz2 + e−iz1
4
i
1 h i(z1 +z2 )
=
e
+ ei(−z1 +z2 ) + ei(z1 −z2 ) + e−i(z1 +z2 )
4
1
sin(z1 ) sin(z2 ) = − eiz1 − e−iz1 eiz2 − e−iz1
4
i
1h
= − ei(z1 +z2 ) − ei(−z1 +z2 ) − ei(z1 −z2 ) + e−i(z1 +z2 ) .
4
cos(z1 ) cos(z2 ) =
(47)
(48)
(49)
(50)
Thus,
cos(z1 ) cos(z2 ) − sin(z1 ) sin(z2 ) =
i
1 h i(z1 +z2 )
e
+ e−i(z1 +z2 ) = cos(z1 + z2 ).
2
(51)
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MH2801: Complex Methods for the Sciences
Y. D. Chong (2016)
As a bonus, these addition formulas now hold for complex inputs as well, not just real inputs.
Higher-order trigonometric addition formulas can be derived in a similar way.
3.5.3
Hyperbolic functions
Euler’s formula also provides us with a link between the trionometric and hyperbolic functions. From the definition of the hyperbolic functions:
sinh(z) =
1 z
e − e−z ,
2
cosh(z) =
1 z
e + e−z
2
(52)
Compare this to our above definition of the complex trigonometric functions:
sin(z) = −
1 iz
i iz
e − e−iz , cos(z) =
e + e−iz
2
2
(53)
From this, we can see that the trigonometric and hyperbolic functions are related by
sin(z) = −i sinh(iz),
sinh(z) = −i sin(iz),
cos(z) = cosh(iz)
cosh(z) = cos(iz)
(54)
(55)
Using these relations, we can relate the addition formulas for trignometric formulas to the
addition formulas for hyperbolic functions, e.g.
cosh(z1 + z2 ) = cos(iz1 + iz2 )
3.6
(56)
= cos(iz1 ) cos(iz2 ) − sin(iz1 ) sin(iz2 )
(57)
= cosh(z1 ) cosh(z2 ) + sinh(z1 ) sinh(z2 ).
(58)
Trajectories in the complex plane
If we have a function z(t) that takes a real input t and gives a complex output z, it is often
useful to plot a curve in the complex plane, called the “parametric trajectory” of z. Each
point on the curve gives the value of z at some t. We will give a few examples below.
First, consider
z(t) = eiωt , ω ∈ R.
(59)
The trajectory is a circle in the complex plane, centered at the origin and with radius 1:
To see why, observe that the function has the form z(t) = r(t) eiθ(t) , where the magnitude
(i.e., the distance from the origin) r(t) = 1 is a constant, and the argument θ(t) = ωt varies
proportionally with t. If ω is positive, the argument increases with t, so the trajectory is
counter-clockwise, as shown in the above figure. If ω is negative, the trajectory is clockwise.
Next, consider
z(t) = e(γ+iω)t , where γ, ω ∈ R.
(60)
For γ = 0, this reduces to the previous example. For γ 6= 0, the trajectory is a spiral:
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MH2801: Complex Methods for the Sciences
Y. D. Chong (2016)
To see why, observe that this function can be written in the form
z(t) = r(t) eiθ(t) ,
(61)
where r(t) = eγt and θ = ωt. Similar to the previous example, the argument varies proportionally with t, so the trajectory loops around the origin. What’s different is that the
magnitude (i.e., the distance from the origin) now either increases or decreases exponentially
with t, depending on the sign of γ. If γ and ω are both positive, then the trajectory is an
anticlockwise spiral moving outwards from the origin. You should work out for yourself how
and why the trajectory behaves if we flip the signs of γ and/or ω.
Finally, consider
1
, α, β ∈ R.
(62)
z(t) =
αt + β
This trajectory is a circle which passes through the origin, as shown below:
The proof is left as an exercise. This is something called a Möbius transformation.
3.7
Why complex numbers? (Optional)
You may wonder what makes the square root of −1 an interesting mathematical object, such
that it’s used to define a whole new system of “complex numbers”. You may also wonder
whether complex numbers are the only generalization of real numbers; could the concept be
usefully extended to even more complicated number systems?
To answer these questions, we observe that complex numbers can be manipulated using
the same rules of algebra as real numbers. We can add, subtract, multiply, and divide (apart
from division by zero) complex numbers, without running into any inconsistencies.
The set of complex numbers C has a special property: it is algebraically closed, meaning
that every complex polynomial equation has solution(s) in C. The set of real numbers,
R, lacks this property, since equations like x2 + 1 = 0 have no solution in R. The “closure” property of C is so important that it’s called the Fundamental Theorem of Algebra.
The Fundamental Theorem implies that C can’t be “algebraically extended” into a more
complicated number system the way we extended R into C.
There do exist extensions of complex numbers that discard one or more of the rules of
algebra. The quaternions are a kind of number system where each quaternionic number
has four real components; these obey a non-commutative algebra where, generally, ab 6=
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MH2801: Complex Methods for the Sciences
Y. D. Chong (2016)
ba. Octonions are eight-component numbers which are both non-commutative and nonassociative: (ab)c 6= a(bc). These and other still-more-complicated number systems have
some applications in physics and other fields, but are overall much less important than C.
Another reason complex numbers are so mathematically rich is that you can do calculus
on them. The study of smooth complex functions is called complex analysis, which will
be discussed later. As we shall see, it is possible to extend the concepts of differentiation
and integration from real functions to complex functions, with many profound outcomes.
By contrast, because quaternions are not commutative, even the concept of “derivative”
becomes tricky to define. Thus, it’s a lot harder to perform mathematical analysis on
quaternions and other more complicated number systems.
3.8
Exercises
1. Let z = x + iy, where x, y ∈ R. For each of the following expressions, find (i) the real
part, (ii) the imaginary part, (iii) the magnitude, and (iv) the complex argument, in
terms of x and y:
•
•
•
•
•
z2
1/z
exp(z)
exp(iz)
cos(z)
2. Show that |z1 z2 | = |z1 | |z2 |, by using (i) the polar representation, and (ii) the Cartesian
representation.
3. Show that (z1 z2 )∗ = z1∗ z2∗ , by using (i) the polar representation, and (ii) the Cartesian
representation.
4. Identify the problem with this chain of equations:
√
√
√ √
−1 = i · i = −1 −1 = −1 · −1 = 1 = 1.
5. With the aid of Euler’s formula, prove that
cos(3x) = 4[cos(x)]3 − 3 cos(x)
(63)
3
sin(3x) = 3 sin(x) − 4[sin(x)] .
(64)
6. For z1 , z2 ∈ C and θ ∈ R, show that Re z1 eiθ + z2 e−iθ = A cos(θ) + B sin(θ), for
some A, B ∈ R. Find explicit expressions for A and B in terms of z1 and z2 .
7. In the complex plane, the conjugation operation corresponds to a reflection about the
real axis. What operation corresponds to a reflection about the imaginary axis?
8. Consider the complex function of a real variable z(t) = 1/(αt + β), where α, β ∈ C and
t ∈ R. For α = 1 and β = i, show that z(t) can be re-expressed as z(s) = (1+eis )/(2i),
where s ∈ (−π, π). Hint: find a real mapping t(s).
Hence, show that the trajectory for arbitrary α, β ∈ C forms a circle.
9. With the help of a computer plotting program, generate complex trajectories for the
following functions (for real inputs t ∈ R). Explain their key features, including the
directions of the trajectories:
i
h
√
exp(it), for β = 10 and for β = 5.
• z(t) = 1 + cos(βt)
2
√
• z(t) = −it ± 1 − t2 .
• z(t) = aeit + be−it , for a = 1, b = −2 and for a = 1, b = 2.
26