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ENGG2420D
Notes for Lecture 4
Inverse functions
Kenneth Shum
23/9/2015
In this lecture we investigate the inverses of elementary functions in complex
analysis.
Function in general. We first review the notion of function in mathematics. A function consists of three things:
1. A set of legitimate inputs to the function, which is usually called the
domain.
2. A set of potential outputs of the function, which is called the range.
3. An association f between the objects in the domain and the objects in
the range, in such a way that each object in the domain is mapped to an
object in the range. We write f (x) = y if x in the domain is associated
with y in the range.
We will use the notations
f :D→R
to indicate that f is a function with domain D and range R.
Example 1. Cardiac signal processing. We want to diagnose whether
a person has a certain heart disease by measuring his/her cardiovascular signal.
We can regard the bio-medical device as a function. An object in the domain is
a cardiovascular signal. The domain contains all possible cardiovascular signals.
The range contains only two objects, namely “positive” and “negative”. The
cardiac signal processing is a function from the domain of all cardiovascular
signals to the set {“positive”, “negative”}.
Forward problem: given an object x in the domain, compute the
corresponding function value f (x).
Inverse problem: given an object y in the range, find all objects
x in the domain such that f (x) = y.
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The solution to the forward problem is always unique, but in general, there may
be none or more than one solutions to the inverse problem.
Two more definitions:
A function f is called one-to-one, or injective, if no two objects
in the domain are mapped to the same object in the range.
A function f is called onto, or surjective, if for any object y in the
range of f , we can find at least one object x in the domain which
are mapped to y.
Example 2. The real cosine function. The domain and range of the
real cosine function are the set of real numbers R, 1
cos : R → R.
√
It is an elementary fact that cos(π/4) = 22 . If we ask the inverse problem “find
√
all angles θ such that cos(θ) = 2/2”, then there are infinitely many solutions,
and they are
π
± + 2πk, for k = 0, ±1, ±2, . . . .
(1)
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However, if we ask the question “find all angles θ such that cos(θ) = 2”,
then there are no solution.
Example 3. The modulus of a complex number. The modulus, or
absolute value, is a function from C to R,
| · | : C → R.
Given a non-negative real number r, there are infinitely many complex numbers
z such that |z| = r. They form a circle centered at the origin with radius r.
If r is a negative real number, then we cannot find any complex number z
whose absolute value is equal to r.
Example 4. The complex exponential function. The complex exponential function
exp : C → C,
is defined by
exp(x + iy) , ex (cos(y) + i sin(y)).
For a given non-zero complex number w = reiθ , there are infinitely many complex numbers z such that ez = w, and they can be written in the form
log(r) + i(θ + 2πk), for k = 0, ±1, ±2, . . . .2
1 We shall use the standard notations that R stands for the the set of all real numbers, and
C stands for the set of all complex numbers.
2 In ENGG2420D, the base of the log function is e by default.
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This is the multi-valued complex log “function”. If we want to avoid multiple
solutions, we can require that the imaginary part to be within the range (−π, π].
This is called the principal value of the complex log function and is commonly
written as Log.
Numerical example: Compute the complex log of −e2 .
Solution. Write −e2 in polar form as e2 eiπ . The complex log of −e2 can be
any one of the following complex numbers,
. . . , 2 − 5πi, 2 − 3πi, 2 − πi, 2 + πi, 2 + 3πi, 2 + 5πi, . . . .
The principal value is 2 + πi, and we can write it as
Log(−e2 ) = 2 + πi.
If w = 0, there is no complex number z such that ez = 0. The complex
exponential function is not onto. The complex log of zero, log(0), is undefined.
Example 5. The complex cosine function. The complex cosine function
cos : C → C,
is defined as
eiz + e−iz
.
2
Given a complex number w, there are infinitely many complex numbers z such
that cos(z) = w. The procedure for computing the inverse of the complex cosine
involves the complex log function. We illustrate this by a numerical example.
Numerical example: Find all complex numbers z such that cos(z) = 2.
Solution. We want to find z such that
cos(z) ,
eiz + e−iz
= 2.
2
For notational convenience, we let u = eiz , and write the above equation as
u+
1
= 4.
u
This can be transformed to a quadratic equation in u,
u2 − 4u + 1 = 0.
The solutions to this quadratic equation are
√
√
2 + 3 and 2 − 3.
So, we need to find all complex numbers z which satisfy
√
√
eiz = 2 + 3 or eiz = 2 − 3.
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√
For the first case, the solutions are the complex log of 2 + 3 divided by i,
namely
√
−i log(2 + 3) + 2πk, for k = 0, ±1, ±2, . . . .
√
For the second case, the solutions are the complex log of 2 − 3 divided by i,
namely
√
−i log(2 − 3) + 2πk, for k = 0, ±1, ±2, . . . .
√
√
√
Finally, we observe that 2−1√3 = 2 + 3, hence log(2 + 3) = − log(2 − 3).
We can summarize the solutions as
√
±i log(2 + 3) + 2πk, for k = 0, ±1, ±2, . . . .
(2)
This conforms with the fact that the cosine function is an even function and is
periodic with period 2π.3 You can also see the similarity between Equations (1)
and (2).
Example 6. The n-th power of a complex number. For a fixed positive
integer n, the n-th power of a complex number is a function from C to C. There
are n solutions to the inverse problem. Given a complex number w = reiθ ,
the complex numbers z such that z n is equal to w are precisely the following n
complex numbers,
r1/n eiθ/n e2πik/n , for k = 0, 1, 2, . . . , n − 1.
The solution r1/n eiθ/n is called the principal root by convention.
Numerical example: Find all 6-th roots of −2π.
Solution. Write −2π in polar form as −2π = 2πeπi . Let ξ be the 6-th root
of unity,
ξ , eπi/3 = cos(π/3) + i sin(π/3).
The 6-th roots of −2π are
√
√
√
√
√
√
6
6
6
6
6
6
2πeπi/6 , 2πeπi/6 ξ, 2πeπi/6 ξ 2 , 2πeπi/6 ξ 3 , 2πeπi/6 ξ 4 , 2πeπi/6 ξ 5 .
√
The principal root is 6 2πeπi/6 .
Remarks:
1. The principal value of the log function and the principal root when taking
the n-th roots are convention which pick one of the many solutions as
the default value.
2. Two functions are regarded as identical function when and only when
they have the same domain, the same range and the same association
of objects. The real cosine function in Example 2 is not the same as
the complex function in Example 5, even though they share the common
notation “cos”. It is because the corresponding domains and ranges are
different. Furthermore, cos−1 (2) has no meaning for real cosine function,
but has infinitely many solutions for the complex cosine function.
3 A function f is said to be even if f (z) = f (−z) for all z in the domain of f . A function
f is 2π-periodic if f (z + 2π) = f (z) for all z in the domain of f .
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