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2.2. MAGIC WITH COMPLEX EXPONENTIALS 97
2.2 Magic with complex exponentials
We don’t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or less from scratch. Feedback will help us to help you, so let us know what you do and don’t understand. Also, if something is not immediately clear you should work through exam- ples ... as usual. The introduction to square roots in school often makes the point that the square root of a negative number is not defined, since after all when we square a number we always get something positive. Then at some point you are told about imaginary numbers, where the basic object is i =
It is not clear, perhaps, whether this is some sort of joke (calling them “imaginary” probably doesn’t help!). Here we are asking you to take these things very seriously. Remember that when you first learned about negative numbers (a long time ago ... ) there was some mystery about what you do when you add, multiply, etc.. In the end the answer is that the rules are the same, and you have to apply them in a consistent way. This is true also for complex or imaginary numbers. We begin by recalling that with x and y real numbers, we can form the complex number z = x + iy. The object i is the square root of negative one, i =
−1. Then if we have two of these numbers z 1 = x 1 + iy 1 (2.45) z 2 = x 2 + iy 2 (2.46)
we can go through all the usual operations of arithmetic:
z 1 + z 2 ≡ (x 1 + iy 1 ) + (x 2 + iy 2 ) (2.47) = (x 1 + x 2 ) + i(y 1 + y 2 ); (2.48) z 1 − z 2 ≡ (x 1 + iy 1 ) − (x 2 + iy 2 ) (2.49) = (x 1 − x 2 ) + i(y 1 − y 2 ); (2.50) (z 1 ) × (z 2 ) ≡ (x 1 + iy 1 ) × (x 2 + iy 2 ) (2.51) = x 1 x 2 + x 1 (iy 2 ) + iy 1 x 2 + iy 1 (iy 2 ) (2.52) = x 1 x 2 + i(x 1 y 2 + x 2 y 1 ) + (i^2 )y 1 y 2 (2.53) = x 1 x 2 + i(x 1 y 2 + x 2 y 1 ) − y 1 y 2 (2.54) = x 1 x 2 − y 1 y 2 + i(x 1 y 2 + x 2 y 1 ), (2.55)
where in the second to last step we use the fact that i^2 = −1. Note that this list leaves out division, which we’ll get back to in a moment.
98 CHAPTER 2. RESONANCE AND RESPONSE
One very useful operation that is new for complex numbers is called “taking the complex conjugate,” or “complex conjugation.” For every com- plex number z = x + iy, the complex conjugate is defined to be z ∗^ = x − iy. Note that in elementary physics we usually use z ∗^ to denote the complex conjugate of z; in the math department and in some more sophisticated physics problems it is conventional to write the complex conjugate of z as ¯z, but of course this is just notation. The crucial fact is that
z × z ∗^ ≡ (x + iy) × (x − iy) (2.56) = x^2 + x(−iy) + iyx + (i)(−i)y 2 (2.57) = x^2 + i(−xy + yx) − (i^2 )y 2 (2.58) = x^2 + y 2. (2.59)
Often we write zz ∗^ = |z|^2 , just the way we write the length of a vector in terms of its dot product with itself, !x·!x = |!x|^2. This is an important thing on its own, as we will see, but also it makes division a lot easier, which we do now. There is a trick, which is to clear the complex numbers from the denom- inator any time we divide:
z (^1) z (^2)
(x 1 + iy 1 ) (x 2 + iy 2 )
z (^1) z (^2)
z ∗ 2 z ∗ 2
z 1 z ∗ 2 z 2 z ∗ 2
z 1 z ∗ 2 |z 2 |^2
(x 1 x 2 + y 1 y 2 ) + i(y 1 x 2 − x 1 y 2 ) x^22 + y 22
Problem 29: You should be able to add, subtract, multiply and divide these pairs of complex numbers: (a) z 1 = 3 + 4i, z 2 = 4 + 3i. (b) z 1 = 3 + 4i, z 2 = 4 − 3 i. (c) z 1 = 7 − 9 i, z 2 = 27 + 12i. And you should be able to make up your own examples!
100 CHAPTER 2. RESONANCE AND RESPONSE
By the same reasoning, one finds
z (^1) z (^2)
|z 1 | |z 2 | [cos(θ 1 − θ 2 ) + i sin(θ 1 − θ 2 )]. (2.75)
Problem 30: Derive Eq (2.75). Also, in terms of these θ 1 and θ 2 , what is the condition for multiplying two complex numbers and getting a real answer?
We now have enough tools to figure out what we mean by the exponential of a complex number. Specifically, let’s ask what we mean by eiφ^. This is a complex number, but it’s also an exponential and so it has to obey all the rules for the exponentials. In particular,
eiφ^1 eiφ^2 = ei(φ^1 +φ^2 )^ (2.76) eiφ^1 eiφ^2 = ei(φ^1 −φ^2 )^. (2.77)
You see that the variable φ behaves just like the angle θ in the geometrial representation of complex numbers. Furthermore, if we take the complex number z = eiφ^ and multiply by its complex conjugate ...
eiφ^ ×
[
eiφ^
] ∗
= eiφ^ ×
[
e−iφ^
]
= ei(φ−φ)^ = 1. (2.78)
Thus z = eiφ^ is a complex number with unit magnitude, and the angle in the complex plane is just φ itself. Thus we see that
eiφ^ = cos φ + i sin φ, (2.79)
which finally tells us what we mean by the complex exponential. Notice that something like this formula had to be true because we know that the solution to the differential equation for the harmonic oscillator can be written either in terms of sines and cosines or in terms of complex exponentials; since solutions are unique, these must be ralted to each other. If you consider the special case of φ = π, then sin φ = 0 and cos φ = −1, leading to the famous Euler formula
eiπ^ + 1 = 0. (2.80)
2.2. MAGIC WITH COMPLEX EXPONENTIALS 101
This is a really beautiful equation, linking the mysterious transcendental numbers e and π with the imaginary numbers.
Problem 31: Derive the sum and difference angle identities by multiplying and dividing the complex exponentials. Use the same trick to derive an expression for cos(3θ) in terms of sin θ and cos θ.
Armed with these tools, let’s get back to our (complex) expression for the trajectory,
x(t) = A exp(+iωt) + B exp(−iωt). (2.81)
We now know that
exp(±iωt) = cos(ωt) ± i sin(ωt), (2.82)
so at least it’s clear what our expression means. To really solve the problem we need to match the initial conditions. We can see that
x(0) = A exp(iω · 0) + B exp(−iω · 0) (2.83) = A + B, (2.84)
because e^0 = 1, as always. Now in principle A and B are complex numbers,
A = ReA + iImA (2.85) B = ReB + iImB, (2.86)
while of course x(0) is the actual position of an object and thus has to be a real number. Let’s substitute and see how this works:
x(0) = A + B = ReA + iImA + ReB + iImB (2.87) = (ReA + ReB) + i(ImA + ImB). (2.88)
So we can match the reality of the initial condition (never mind its value!) only if
ImB = −ImA. (2.89)
2.2. MAGIC WITH COMPLEX EXPONENTIALS 103
Now x(t) is the sum of a complex number and its complex conjugate. But when we add a complex number to its complex conjugate, we cancel the imaginary part and double the real part:
z + z ∗^ = [Re(z) + iIm(z)] + [Re(z) − iIm(z)] (2.101) = 2Re(z). (2.102)
Thus x(t), according to Eq (2.100) will be real at all times. This is good, of course (!). Interestingly, we didn’t actually use this condition—all we did was to be sure that we match the initial conditions, which of course are real. This is sufficient to insure that trajectories are real forever, which is nice. To proceed further, we recall that all complex numbers can be written as
z = Rez + iImz (2.103) = |z|(cos φ + i sin φ) (2.104) = |z| exp(iφ), (2.105)
where z is the magnitude of the complex number and φ is its phase,
|z| =
[Re(z)]^2 + [Im(z)]^2 (2.106)
φ = tan−^1
[
Im(z) Re(z)
]
If we do this rewriting of A,
A = |A| exp(iφA ), (2.108)
then the trajectory becomes
x(t) = [|A| exp(iφA ) exp(+iωt)] + [|A| exp(iφA ) exp(+iωt)]∗^ (2.109) = [|A| exp(+iωt + iφA )] + [|A| exp(+iωt + iφA )]∗^. (2.110)
Thus
x(t) = [|A| exp(+iωt + iφA )] + [|A| exp(+iωt + iφA )]∗ = 2Re[|A| exp(+iωt + iφA )] (2.111) = 2 |A|Re[exp(+iωt + iφA )] (2.112) = 2 |A| cos(ωt + φA ). (2.113)
So, we are done, except that we have to connect this solution to the initial conditions.
104 CHAPTER 2. RESONANCE AND RESPONSE
We want to find the arbitrary parameters |A| and φA in terms of the initial position and initial velocity. Let’s just calculate:
x(t = 0) = 2 |A| cos(ω(0) + φA ) (2.114) = 2 |A| cos φA (2.115)
v(t = 0) =
dx(t) dt
t=
d dt [2|A| cos(ωt + φA )]
t=
= − 2 |A|ω sin(ωt + φA )
t=
v(t = 0) = − 2 |A|ω sin φA (2.119) v(t = 0) ω = − 2 |A| sin φA. (2.120)
So we have two equations, Eq’s (2.115) and (2.120), that link our parameters to the initial conditions. To solve these equations, note that if we sum the square of the two equations we have
[x(0)]^2 +
[
v(0) ω
] 2
= (2|A|)^2 cos 2 φA + (2|A|)^2 sin^2 φA (2.121)
= (2|A|)^2 , (2.122)
so that
2 |A| =
[x(0)]^2 +
[
v(0) ω
] 2
Similarly, if we take Eq (2.120) and divide by Eq (2.115), we find
v(0) ωx(0)
2 |A| sin φA 2 |A| cos φA
= tan φA , (2.125)
and hence
φA = tan−^1
[
v(0) ωx(0)
]
Thus the amplitude |A| of the oscillation is related to the initial position, with an extra contribution from the initial velocity, while the phase depends
106 CHAPTER 2. RESONANCE AND RESPONSE
