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MATHEMATICS II
I B.Sc ELECTRONICS AND COMMUNICATION
SYSTEMS
UNIT I
DIFFERENTIAL EQUATIONS
2nd-order differential equations - homogeneous and with
constant coefficients
Second-order differential equations, by definition, contain a second derivative, like d2y/dx2, for
example. As well as the second derivative, there may also be a first derivative in the equation
and sometimes a term involving just y itself.
The second-order differential equations we will look at have "constant coefficients", which
means that these three possible terms, that is the second derivative, the first derivative and the yterm, all just have a number in front of them (their coefficient), rather than a function of x.
That means that the equations we're looking at have this general form:
where a, b and c are just numbers and f(x) is any function of x.
In this section, we will look at equations which have zero right-hand side, i.e. for which the
function f(x) is zero. Such equations are called homogeneous. In the next section we'll look at
inhomogeneous equations in which f(x) is not zero.
The first fact to take in about constant-coefficient differential equations is this:
If y1(x) is a solution to the equation and y2(x) is also a solution, then the function formed by
adding the two, i.e. y3(x)=y1(x)+y2(x) is also a solution.
Here's the general equation again:
Now, because y1 satisfies the equation ,that means that we can replace y by y1 and the equation
will be true, so:
Similarly for y2:
Now we can always add two equations together and the resulting equation will also hold true, so
let's add those two together. Gathering up similar terms into brackets, we get:
But adding the second derivative of y1 to the second derivative of y2 gives the same result as
we'd get if we added y1 to y2 first, and then differentiated twice. So this equation is also true:
Now we can see that if y3 is defined by y3=y1+y2, so that we can replace every occurrence of that
sum y1+y2 by y3, then this equation holds:
And that means that y3 satisfies the equation, it is a solution.
The second fact is this:
The general solution to homogeneous, constant-coefficient, 2nd-order differential equations is
always of the form:
where A, B, m1, m2 are all constants.
Why is this the general solution:
Here's the general homogeneous equation again:
?
To get an idea of what the solution might be like, let's first try the case a=0, so the secondderivative term disappears. Then we're left with:
Now that's just a first-order differential equation and we can separate the variables to solve it (see
earlier sections). The solution is:
If we now define a constant m as -c/b, just to make it neater, we get:
That's the solution to the 2nd-order equation if a=0. That suggests we try that form for y(x) in
the 2nd-order equation for non-zero a.
So, if y(x)=Aemx, then
and
.
and substituting these into the differential equation we get:
Now we can take the Aemx factor outside because they've all got it, giving us:
The expression Aemx is not zero in general, which means that the quadratic expression
am2+bm+c must be zero for this equation to hold.
And if this equation holds then y(x)=Aemx is a solution to our 2nd-order equation, even when a is
not zero.
So, as long as the constant m satisfies the quadratic equation below, y(x)=Aemx is a solution to
the differential equation. This quadratic equation is called the auxiliary equation:
Notice that if a=0 we get back to the solution m=-c/b that we had above.
But, for non-zero a, quadratic equations generally give us two roots, so we can expect to find
two possible values for m.
This is where we can use that "first fact", that told us we could add two solutions and the result
would also be a solution: if we denote the two possible values of m by m1 and m2, we get the
following solution for y(x), as given earlier:
The other two constants A and B are the two constants of integration (remember there's two now
because it's a second-order equation) and they are found only when we fix some boundary
conditions. This is just like the constant "c" that we usually had in our solutions to first-order
equations.
Here's an example, using the above.
What is the solution of this equation:
Here are the steps to follow:
(i)Write down the general solution (fact 2 above)
(ii)Use the auxiliary equation to find the constants m1 and m2
That's it!
The solution of
The first step is to write down the general solution for y(x):
Next we need to find m1 and m2, using the auxiliary equation. The coefficients are just those
from the differential equation above:
We can find the roots of this quadratic equation using the quadratic formula, which gives us:
The two possible values for m are therefore:
and
So the solution for y(x) is:
Here's another example, find the general solution of this equation:
The solution of
The first step is to write down the general solution for y(x):
Next we need to find m1 and m2, using the auxiliary equation. The coefficients are just those
from the differential equation above:
We can find the roots of this quadratic equation using the quadratic formula , which gives us:
The two possible values for m are therefore:
and
So the solution for y(x) is:
Now here's a complication: what happens if when we work out m1 and m2 we find that the
quadratic auxiliary equation has a double root, as in this case:
The solution of
The first step is to write down the general solution for y(x):
Next we need to find m1 and m2, using the auxiliary equation. The coefficients are just those
from the differential equation above:
As before we can find the roots of this quadratic equation using the quadratic formula, which
gives us:
There is only one possible value for m in this case:
So one solution is y1(x)=Ae-x, but what about the second solution?
In such cases as these, there is a second solution given by y2(x)=Bxe-x. Notice there is an "x" in
front of the exponential in this second solution and the exponent is the same as the first solution,
"-x".
So, combining those two solutions, we get the solution for y(x) as:
Now here's another complication: what happens if the values you get for m1 and m2 are complex,
as in this case:
Now here's another complication: what happens if the values you get for m1 and m2 are complex,
as in this case:
The solution of
The first step is to write down the general solution for y(x):
Next we need to find m1 and m2, using the auxiliary equation. The coefficients are just those
from the differential equation above:
As before we can find the roots of this quadratic equation using the quadratic formula, which
gives us:
There are two complex values for m in this case:
and
So the solution is
.
This is more easily understood, however, if we remember from the what we mean by the
expression eix, namely:
eix=cos(x)+isin(x) and e-ix=cos(x)-isin(x).
Using that we can rewrite the above solution for y(x) as:
where the constants C and D are related to the constants A and B by
This shows that solutions to 2nd-order, homogeneous, constant-coefficient differential equations
can be sinusoidal in character as well as exponential.
Summary of results
The auxiliary equation can either give us two real roots, a double root, or two complex roots for
m. The table below shows the solutions to the differential equations in each of those possible
cases.
Nature of auxiliary equation
Solution to differential equation
Two real roots, m=m1 and m=m2
A double root, m=m1
Two complex roots, m=mR+imI and m=mR-imI
UNIT III
FOURIER TRANSFORMS
Definition
The Fourier Transform is merely a restatement of the Fourier Integral:
.
Using the complex form of Cosine, we can easily prove that the above integral can be re-written
as:
.
The above integral can be expressed by the following Fourier Transform pair:
Since is a dummy variable, we can replace it with
and its inverse transform as:
where
and
and define the Fourier transform of
are the Fourier and its inverse transform operators, respectively.
Fourier Cosine and Sine Transforms
If
is an even function, then its Fourier Integral is equivalent to the following pair of
equations:
where
is called the Fourier Cosine Transform operator.
On the other hand, if
is an odd function, then its Fourier Integral is equivalent to the
following pair of equations:
where
is called the Fourier Sine Transform operator.
Properties of Fourier Transform
The properties of the Fourier transform are summarized below. The properties of the Fourier
expansion of periodic functions discussed above are special cases of those listed here. In the
following, we assume

Linearity

Time shift
Proof: Let

Frequency shift
and
, i.e.,
, we have
.
Proof: Let

, i.e.,
Time reversal
Proof:
Replacing
by
, we get
, we have

Even and Odd Signals and Spectra
If the signal
is an even (or odd) function of time, its spectrum
(or odd) function of frequency:
is an even
 and
Proof: If
is even, then according to the time reversal property, we have
i.e., the spectrum
have
is also even. Similarly, if
i.e., the spectrum
 Time and frequency scaling
Proof: Let
, i.e.,
is also odd.
, where
is a scaling factor, we have
is odd, we
Note that when
,
, time function
is compressed and
transform, i.e., compressing one of the
is stretched, and
is compressed; when
is stretched. This is a general feature of Fourier
and
will stretch the other and vice versa. In
particular, when
,
is stretched to approach a constant, and
compressed with its value increased to approach an impulse; on the other hand, when
is compressed with its value increased to approach an impulse and
stretched to approach a constant.
 Complex Conjugation
Proof: Taking the complex conjugate of the inverse Fourier transform, we get
Replacing
by
we get the desired result:
We further consider two special cases:


If
is real, then
is
,
is

i.e., the real part of the spectrum is even (with respect to frequency
imaginary part is odd:



), and the
If
is imaginary, then


i.e., the real part of the spectrum is odd, and the imaginary part is even:


If the time signal
is one of the four combinations shown in the table (real even, real odd,
imaginary even, and imaginary odd), then its spectrum
table entry:

if
is real
even,
if
if
odd
is imaginary
odd,
even
is Even
and
if
is given in the corresponding
even
is Odd
,
even
,
even
and
odd
,
odd
,
odd
Note that if a real or imaginary part in the table is required to be both even and odd at the same
time, it has to be zero.
These properties are summarized below:
1
2
3
4
5
6
real
even
, odd
real and even
real and even
real and odd
imaginary and odd
imaginary
odd
imaginary and even
imaginary and even
imaginary and odd
real and odd
, even
As any signal can be expressed as the sum of its even and odd components, the first three items
above indicate that the spectrum of the even part of a real signal is real and even, and the
spectrum of the odd part of the signal is imaginary and odd.
 Symmetry (or Duality)
Or in a more symmetric form:
Proof: As
, we have
Letting
, we get
Interchanging
and
we get:
or
In particular, if the signal is even:
 In particular, if the signal is even:
then we have
For example, the spectrum of an even square wave is a sinc function, and the spectrum of a sinc
function is an even square wave.
 Multiplication theorem
Proof:
 Parseval's equation
In the special case when
Parseval 1799):
, the above becomes the Parseval's equation (Antoine
where
is the energy density function representing how the signal's energy is distributed along the frequency
axes. The total energy contained in the signal is obtained by integrating
frequency axes.
over the entire
The Parseval's equation indicates that the energy or information contained in the signal is
reserved, i.e., the signal is represented equivalently in either the time or frequency domain with
no energy gained or lost.
 Correlation
The cross-correlation of two real signals
Specially, when
is defined as
, the above becomes the auto-correlation of signal
Assuming
, we have
multiplication theorem,
can be written as
i.e.,
and
and according to
that is, the auto-correlation and the energy density function of a signal
pair.
are a Fourier transform
 Convolution Theorems
 The convolution theorem states that convolution in time domain corresponds to multiplication
in frequency domain and vice versa:
Proof of (a):
Proof of (b):
 Time Derivative
Proof: Differentiating the inverse Fourier transform
with respect to
 Repeating this process we get
 Time Integration
First consider the Fourier transform of the following two signals:
According to the time derivative property above
we get:
we get
and
Why do the two different functions have the same transform?
In general, any two function
and
with a constant difference
have the
same derivative
, and therefore they have the same transform according the above
method. This problem is obviously caused by the fact that the constant difference is lost in the
derivative operation. To recover this constant difference in time domain, a delta function needs
to be added in frequency domain. Specifically, as function
component, its transform does not contain a delta:
To find the transform of
does not have DC
, consider
and
The added impulse term
directly reflects the constant
in time domain.
Now we show that the Fourier transform of a time integration is
Proof:
First consider the convolution of
and
:
 Due to the convolution theorem, we have
 Frequency Derivative
Proof: We differentiate the Fourier transform of
i.e.,
Multiplying both sides by
, we get
with respect to
to get
Repeating this process we get
UNIT IV
BETA AND GAMMA FUNCTIONS
Beta and Gamma functions
The gamma function { gamma(p) } is defined as
The beta function {beta(m,n)} is defined as
Another definition for the beta integral (with limits 0 to infinity) n by
The relation between the beta and gamma functions is given by
Recurrence relation of gamma functions is given by
Relation between beta funtion and integrals involving trigonometric functions
is given by
Also note that
PROPERTIES OF GAMMA AND BETA FUNCTIONS
Beta distribution
Definition of the Beta distribution
A first example : order statistics of the uniform distribution
Using only very simple arguments, we showed that the distribution of the kth order statistic of the
uniform distribution in [0, 1] is :
where n is the sample size.
It is often convenient to express factorials as Gamma functions, and owing to the property of the
Gamma function :
Γ(m) = (m - 1)!
when m is an integer, the above expression becomes
with :
*α=k
*β=n-k+1
Here, α and β are integers, but the above expression still makes mathematical sense if this
condition is relaxed and if α and β are only constrained to be positive real numbers, but it is then
not clear that fα,β (x) represents a probability density function anymore. This would be the case
only if :
that is if
for all pairs of positive real numbers (α, β).
This is indeed the case as will be shown in the Tutorial below.
Relationship between the Beta function and the Gamma
function
The integral
defines a function of α and β known as the Beta function. So what we will show is that the Beta
and the Gamma functions entertain the following relationship
In what follows, we'll dispense with the Beta function and use this relationship between the Beta
and the Gamma functions to express all results in terms of the Gamma function only.
Gamma(1/2)
We'll show that an immediate consequence of this important equation is :
The Beta distribution
So we have identified a family of probability distributions that are continuous in [0, 1] and
indexed by two real positive parameters α and β. These distributions are collectively called the
Beta distribution, that will be denoted Beta(α, β).
The probability density function of the Beta distribution is :
PROBLEMS
1.
UNIT V
COMPLEX NUMBERS
Complex number
From Wikipedia, the free encyclopedia
Jump to: navigation, search
A complex number can be visually represented as a pair of numbers forming a vector on a diagram
called an Argand diagram, representing the complex plane.
A complex number is a number consisting of a real and imaginary part. It can be written in the
form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the
property i 2 = −1.[1] The complex numbers contain the ordinary real numbers, but extend them by
adding in extra numbers and correspondingly expanding the understanding of addition and
multiplication.
Complex numbers were first conceived and defined by the Italian mathematician Gerolamo
Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations.[2]
The solution of a general cubic equation in radicals (without trigonometric functions) may
require intermediate calculations containing the square roots of negative numbers, even when the
final solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to
the fundamental theorem of algebra, which shows that with complex numbers, a solution exists
to every polynomial equation of degree one or higher. Complex numbers thus form an
algebraically closed field, where any polynomial equation has a root.
The rules for addition, subtraction, multiplication, and division of complex numbers were
developed by the Italian mathematician Rafael Bombelli.[3] A more abstract formalism for the
complex numbers was further developed by the Irish mathematician William Rowan Hamilton,
who extended this abstraction to the theory of quaternions.
Complex numbers are used in a number of fields, including: engineering, electromagnetism,
quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers
for a mathematical construct is the field of complex numbers, the name usually reflects that fact.
Examples are complex analysis, complex matrix, complex polynomial, and complex Lie algebra.
Complex numbers are plotted on the complex plane, on which the real part is on the horizontal
axis, and the imaginary part on the vertical axis.
Contents
[hide]











1 Definitions and basic properties
o 1.1 Notation
o 1.2 Equality
o 1.3 Operations
o 1.4 Absolute value and distance
o 1.5 Conjugation
o 1.6 Formal development
o 1.7 Elementary functions
o 1.8 Exponentiation
2 The complex plane
o 2.1 Geometric interpretation of the operations
o 2.2 Polar form
 2.2.1 Operations in polar form
3 Some advanced properties
o 3.1 Matrix representation of complex numbers
o 3.2 Real vector space
o 3.3 Solutions of polynomial equations
o 3.4 Construction and algebraic characterization
o 3.5 Characterization as a topological field
4 Complex analysis
5 Applications
o 5.1 Control theory
o 5.2 Signal analysis
o 5.3 Improper integrals
o 5.4 Quantum mechanics
o 5.5 Relativity
o 5.6 Applied mathematics
o 5.7 Fluid dynamics
o 5.8 Fractals
6 History
7 See also
8 Notes
9 References
o 9.1 Mathematical references
o 9.2 Historical references
10 Further reading
11 External links
Definitions and basic properties
Notation
The set of all complex numbers is usually denoted by C, or in blackboard bold by
.
Although other notations can be used, complex numbers are usually written in the form
where a and b are real numbers, and i is the imaginary unit, which has the property i 2 = −1. The
real number a is called the real part of the complex number, and the real number b is the
imaginary part.[4] For example, 3 + 2i is a complex number, with real part 3 and imaginary part
2. If z = a + bi, the real part a is denoted Re(z) or ℜ(z), and the imaginary part b is denoted Im(z)
or ℑ(z).
The complex numbers (C) are regarded as an extension of the real numbers (R) by considering
every real number as a complex number with an imaginary part of zero. The real number a is
identified with the complex number a + 0i. Complex numbers with a real part of zero (Re(z)=0)
are called imaginary numbers. Instead of writing 0 + bi, that imaginary number is usually
denoted as just bi. If b equals 1, instead of using 0 + 1i or 1i, the number is denoted as i.
In some disciplines (in particular, electrical engineering, where i is a symbol for current), the
imaginary unit i is instead written as j, so complex numbers are sometimes written as a + bj or a
+ jb.
Equality
Two complex numbers are said to be equal if and only if their real parts are equal and their
imaginary parts are equal. In other words, if the two complex numbers are written as a + bi and
c + di with a, b, c, and d real, then they are equal if and only if a = c and b = d.
Operations
Complex numbers are added, subtracted, multiplied, and divided by formally applying the
associative, commutative and distributive laws of algebra, together with the equation i 2 = −1:

Addition:


Subtraction:
Multiplication:

Division:
where c and d are not both zero. This is obtained by multiplying both the numerator and the
denominator by the conjugate of the denominator c + di, which is (c − di).
Absolute value and distance
The absolute value (or modulus or magnitude) of a complex number
In polar form, described below,
has three important properties:
where
it is
is
The absolute value
if and only if
(triangle inequality)
for all complex numbers z and w. These imply that |1| = 1 and |z/w| = |z|/|w|. By defining the
distance function d(z, w) = |z − w|, we turn the set of complex numbers into a metric space and
we can therefore talk about limits and continuity.
Conjugation
Geometric representation of z and its conjugate
in the complex plane
The complex conjugate of the complex number z = x + yi is defined to be x − yi, written as or
. As seen in the figure, is the "reflection" of z about the real axis, and so both
and
are real numbers. Many identities relate complex numbers and their conjugates.
Conjugating twice gives the original complex number:
The square of the absolute value is obtained by multiplying a complex number by its conjugate:
if z is non-zero.
The latter formula is the method of choice to compute the multiplicative inverse of a complex
number if it is given in rectangular coordinates.
Conjugation distributes over the standard arithmetic operations:
That conjugation distributes over all the algebraic operations and many functions, e.g.
is rooted in the ambiguity in choice of i (−1 has two square roots). It is
important to note, however, that the function
holomorphic function).
is not complex-differentiable (see
The real and imaginary parts of a complex number can be extracted using the conjugate:
if and only if z is real
if and only if z is purely imaginary
Formal development
In a rigorous setting, it is not acceptable to simply assume that there exists a number which when
squared gives −1. There are several ways of defining C, building on the base of real numbers.
Firstly, write C for R2, the set of ordered pairs of real numbers, and define operations on
complex numbers in C according to
It is then just a matter of notation to express (a, b) as a + ib. This means we can associate the
numbers (a, 0) with the real numbers, and write i = (0, 1). Since (0, 1)·(0, 1) = (−1, 0), we have
found i by constructing it, not postulating it. Using these formal operations on R2, it is easy to
check that we satisfy the field axioms (associativity, commutativity, identity, inverses,
distributivity). In particular, R is a subfield of C.
Though this low-level construction does accurately describe the structure of the complex
numbers, the definitions seem arbitrary, so secondly C can be considered algebraically.[citation
needed]
In algebra (the theory of group-like structures), this explicit definition of operations in fact
turns out to be the mechanism behind the idea of constructing the algebraic closure of the reals,
that is, adding in some elements to R to make a new field, of which R is a subfield, where every
non-constant polynomial has a root. Finally, yet another way of characterising C is in terms of its
topological properties. Details of these are given below.
Elementary functions
Main article: Elementary function
One of the most important functions on the complex numbers is perhaps the exponential function
exp(z), also written ez, defined in terms of the infinite series
The elementary functions are those which can be finitely built using exp and the arithmetic
operations given above, as well as taking inverses; in particular, the inverse of the exponential
function, the logarithm. The real-valued logarithm over the positive reals is well-defined, and the
complex logarithm generalises this idea. The inverse of exp is shown to be
where arg is the argument defined below, and ln the real logarithm. As arg is a multivalued
function, unique only up to a multiple of 2π, log is also multivalued. The principal value of log is
often taken by restricting the imaginary part to the interval (−π,π].
The familiar trigonometric functions are composed of these, so they are also elementary. For
example,
Hyperbolic functions such as sinh are similarly constructed.
Exponentiation
For more details on this topic, see Exponentiation.
Raising numbers to positive integer powers is the same as repeated multiplication:
Negative integer powers are defined as for real numbers, since 1/zn is the only way of
interpreting z−n such that the familiar rules of indices still work (z−n = z−n(zn/zn) = z−n+n/zn = 1/zn).
Similar considerations show that rational real powers can be defined as for the reals, so z1/n is the
nth root of z. Such roots are not unique and careful treatment of powers is needed; for example
84/3 = (81/3)4 has three possible values, the real 16 and two complex values, as there are three
cube roots of 8.
For arbitrary complex powers zω will generally be multi-valued. To agree with the definitions so
far it can be calculated with
which is the general extension of exponentiation to the complex numbers.
The complex plane
Figure 1: A complex number plotted as a point (red) and position vector (blue) on an Argand diagram; a
+ bi is the rectangular expression of the point.
A complex number can be viewed as a point or position vector in a two-dimensional Cartesian
coordinate system called the complex plane or Argand diagram (see Pedoe 1988 and
Solomentsev 2001), named after Jean-Robert Argand. The numbers are conventionally plotted
using the real part as the horizontal component, and imaginary part as vertical (see Figure 1).
These two values used to identify a given complex number are therefore called its Cartesian-,
rectangular-, or algebraic form.
Geometric interpretation of the operations
The operations described algebraically above can be visualised using Argand diagrams.
X = A + B: The sum of two points A and B of the complex plane
is the point X = A + B such that the triangles with vertices 0, A,
B, and X, B, A, are congruent. Thus the addition of two
complex numbers is the same as vector addition of two
vectors.
X = AB: The product of two points A and B is the point X = AB
such that the triangles with vertices 0, 1, A, and 0, B, X, are
similar.
X = A*: The complex conjugate of a point A is the point X = A*
such that the triangles with vertices 0, 1, A, and 0, 1, X, are
mirror images of each other.
These geometric interpretations allow problems of algebra to be translated into geometry. And,
conversely, geometric problems can be examined algebraically. For example, the problem of the
geometric construction of the 17-gon was by Gauss translated into the analysis of the algebraic
equation x17 = 1 (see Heptadecagon).
Polar form
For more details on this topic, see Polar coordinate system.
Figure 2: The argument φ and modulus r locate a point on an Argand diagram; r(cosφ + isinφ) or reiφ are
polar expressions of the point.
The diagrams suggest various properties. Firstly, the distance of a point z from the origin (shown
as r in Figure 2) is known as the modulus, absolute value, or magnitude, and written | z | . By
Pythagoras' theorem,
In general, distances between complex numbers are given by the distance function d(z,w) = | z −
w | , which turns the complex numbers into a metric space and introduces the ideas of limits and
continuity. All of the standard properties of two dimensional space therefore hold for the
complex numbers, including important properties of the modulus such as non-negativity, and the
triangle inequality (
for all z, w).
Secondly, the argument or phase of a complex number z = x + yi is the angle to the real axis
(shown as φ in Figure 2), and is written as arg(z). As with the modulus, the argument can be
found from the rectangular form x + iy:
or
(adding π when x < 0 so that x + iy = r(cosφ +
isinφ).
The value of φ can change by any multiple of 2π and still give the same angle (note that radians
are being used). Hence, the arg function is sometimes considered as multivalued, but often the
value is chosen to lie in the interval ( − π,π], or [0,2π) (this is the principal value).
Together, these give another way of representing complex numbers, the polar form, as the
combination of modulus and argument fully specify the position of a point on the plane
(confirmed by recovering the original rectangular co-ordinates
from the polar pair (r,φ)). This can be notated in various ways, including
called trigonometric form, and sometimes abbreviated r cis φ, or using Euler's formula
which is called exponential form. In electronics it is common to use angle notation to represent a
phasor with amplitude A and phase θ as
In angle notation θ may be in either radians or degrees. In electronics it is also common to use j
instead of i, as not to create confusion with the electric current which is usually called i.
Operations in polar form
Multiplication and division have simple formulas in polar form:
and
This form demonstrates that multiplication can be visualised as a simultaneous stretching and
rotation of one of the multiplicands, adding to its angle the phase of the other and scaling its
length. For example, multiplying by i corresponds to a quarter-rotation counter-clockwise, from
which it is clear why i 2 = −1. In particular, multiplication by any number on the unit circle
around the origin is a pure rotation. Division is the same, in reverse.
Exponentiation is also simple; with integer exponents:
[De Moivre's formula]
Arbitrary complex exponents are discussed in Exponentiation.
Finally, polar forms are also useful for finding roots. Any complex number z satisfying zn = c
(for n a positive integer) is called an nth root of c. If c is non-zero, there are exactly n distinct nth
roots of c (by the fundamental theorem of algebra). Let c = re iφ with r > 0; then the set of nth
roots of c is
where
represents the usual (positive) nth root of the positive real number r. If c = 0, then the
only nth root of c is 0 itself, which as nth root of 0 is considered to have multiplicity n, hence
these do represent all the n roots. Note that the roots differ only by the rotations e2kπi/n, the nth
roots of unity, so all the roots of c lie on a circle about the origin.
Some advanced properties
Matrix representation of complex numbers
While usually not useful, alternative representations of the complex field can give some insight
into its nature. One particularly elegant representation interprets each complex number as a 2×2
matrix with real entries which stretches and rotates the points of the plane. Every such matrix has
the form
where a and b are real numbers. The sum and product of two such matrices is again of this form,
and the product operation on matrices of this form is commutative. Every non-zero matrix of this
form is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a
field, isomorphic to the field of complex numbers. Every such matrix can be written as
which suggests that we should identify the real number 1 with the identity matrix
and the imaginary unit i with
a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed
equal to the 2×2 matrix that represents −1.
More formally, this matrix representation is the regular representation of the complex numbers,
thought of as an R-algebra (an R-vector space with a multiplication), with respect to the basis
1,i: the complex numbers are a 2-dimensional vector space over the real numbers, and
multiplication by a complex number is a linear map (by distributivity) of the complex numbers to
themselves, which is thus represented by a 2×2 matrix once a basis has been chosen. Thus this is
not an ad hoc construction, but can be applied to any K-algebra over a field. For example, if the
matrix elements are themselves complex numbers, the resulting algebra is that of the
quaternions; stated alternatively, the quaternions are a 2-dimensional C-algebra, and hence their
regular representation is as 2×2 complex matrices. Generalizing alternatively, this matrix
representation is one way of expressing the Cayley–Dickson construction of algebras.
The square of the absolute value of a complex number expressed as a matrix is equal to the
determinant of that matrix.
If matrix multiplication is viewed as a transformation of the plane, then the transformation
rotates points through an angle equal to the argument of the complex number and scales by a
factor equal to the complex number's absolute value. The conjugate of the complex number z
corresponds to the transformation which rotates through the same angle as z but in the opposite
direction, and scales in the same manner as z; this can be represented by the transpose of the
matrix corresponding to z. This is generalized in the polar decomposition of matrices.
It should also be noted that the two eigenvalues of the 2x2 matrix representing a complex
number are the complex number itself and its conjugate.
While the above is a linear representation of C in the 2 × 2 real matrices, it is not the only one.
Any matrix
has the property that its square is the negative of the identity matrix: J2 = − I. Then
is also isomorphic to the field C, and gives an alternative
complex structure on R2. This is generalized by the notion of a linear complex structure.
Real vector space
C is a two-dimensional real vector space. Unlike the reals, the set of complex numbers cannot be
totally ordered in any way that is compatible with its arithmetic operations: C cannot be turned
into an ordered field. More generally, no field containing a square root of −1 can be ordered.
R-linear maps C → C have the general form
with complex coefficients a and b. Only the first term is C-linear, and only the first term is
holomorphic; the second term is real-differentiable, but does not satisfy the Cauchy-Riemann
equations.
The function
corresponds to rotations combined with scaling, while the function
corresponds to reflections combined with scaling.
Solutions of polynomial equations
A root of the polynomial p is a complex number z such that p(z) = 0. A surprising result in
complex analysis is that all polynomials of degree n with real or complex coefficients have
exactly n complex roots (counting multiple roots according to their multiplicity). This is known
as the fundamental theorem of algebra, and it shows that the complex numbers are an
algebraically closed field. Indeed, the complex numbers are the algebraic closure of the real
numbers, as described below.
Construction and algebraic characterization
One construction of C is as a field extension of the field R of real numbers, in which a root of
x2+1 is added. To construct this extension, begin with the polynomial ring R[x] of the real
numbers in the variable x. Because the polynomial x2+1 is irreducible over R, the quotient ring
R[x]/(x2+1) will be a field. This extension field will contain two square roots of -1; one of them
is selected and denoted i. The set {1, i} will form a basis for the extension field over the reals,
which means that each element of the extension field can be written in the form a+ b·i.
Equivalently, elements of the extension field can be written as ordered pairs (a,b) of real
numbers.
Although only roots of x2+1 were explicitly added, the resulting complex field is actually
algebraically closed &dash; every polynomial with coefficients in C factors into linear
polynomials with coefficients in C. Because each field has only one algebraic closure, up to field
isomorphism, the complex numbers can be characterized as the algebraic closure of the real
numbers.
The field extension does yield the well-known complex plane, but it only characterizes it
algebraically. The field C is characterized up to field isomorphism by the following three
properties:



it has characteristic 0
its transcendence degree over the prime field is the cardinality of the continuum
it is algebraically closed
One consequence of this characterization is that C contains many proper sub fields which are
isomorphic to C (the same is true of R, which contains many sub fields isomorphic to itself[citation
needed]
). As described below, topological considerations are needed to distinguish these subfields
from the fields C and R themselves.
Characterization as a topological field
As just noted, the algebraic characterization of C fails to capture some of its most important
topological properties. These properties are key for the study of complex analysis, where the
complex numbers are studied as a topological field.
The following properties characterize C as a topological field:[citation needed]



C is a field.
C contains a subset P of nonzero elements satisfying:
o P is closed under addition, multiplication and taking inverses.
o If x and y are distinct elements of P, then either x-y or y-x is in P
o If S is any nonempty subset of P, then S+P=x+P for some x in C.
C has a nontrivial involutive automorphism x→x*, fixing P and such that xx* is in P for any
nonzero x in C.
Given a field with these properties, one can define a topology by taking the sets

as a base, where x ranges over the field and p ranges over P.
To see that these properties characterize C as a topological field, one notes that P ∪ {0} ∪ -P is
an ordered Dedekind-complete field and thus can be identified with the real numbers R by a
unique field isomorphism. The last property is easily seen to imply that the Galois group over the
real numbers is of order two, completing the characterization.
Pontryagin has shown that the only connected locally compact topological fields are R and C.
This gives another characterization of C as a topological field, since C can be distinguished from
R by noting that the nonzero complex numbers are connected, while the nonzero real numbers
are not.
Complex analysis
For more details on this topic, see Complex analysis.
The study of functions of a complex variable is known as complex analysis and has enormous
practical use in applied mathematics as well as in other branches of mathematics. Often, the most
natural proofs for statements in real analysis or even number theory employ techniques from
complex analysis (see prime number theorem for an example). Unlike real functions which are
commonly represented as two-dimensional graphs, complex functions have four-dimensional
graphs and may usefully be illustrated by color coding a three-dimensional graph to suggest four
dimensions, or by animating the complex function's dynamic transformation of the complex
plane.
Applications
Some applications of complex numbers are:
Control theory
In control theory, systems are often transformed from the time domain to the frequency domain
using the Laplace transform. The system's poles and zeros are then analyzed in the complex
plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex
plane.
In the root locus method, it is especially important whether the poles and zeros are in the left or
right half planes, i.e. have real part greater than or less than zero. If a system has poles that are



in the right half plane, it will be unstable,
all in the left half plane, it will be stable,
on the imaginary axis, it will have marginal stability.
If a system has zeros in the right half plane, it is a nonminimum phase system.
Signal analysis
Complex numbers are used in signal analysis and other fields for a convenient description for
periodically varying signals. For given real functions representing actual physical quantities,
often in terms of sines and cosines, corresponding complex functions are considered of which the
real parts are the original quantities. For a sine wave of a given frequency, the absolute value |z|
of the corresponding z is the amplitude and the argument arg(z) the phase.
If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions,
these periodic functions are often written as complex valued functions of the form
where ω represents the angular frequency and the complex number z encodes the phase and
amplitude as explained above.
In electrical engineering, the Fourier transform is used to analyze varying voltages and currents.
The treatment of resistors, capacitors, and inductors can then be unified by introducing
imaginary, frequency-dependent resistances for the latter two and combining all three in a single
complex number called the impedance. (Electrical engineers and some physicists use the letter j
for the imaginary unit since i is typically reserved for varying currents and may come into
conflict with i.) This approach is called phasor calculus. This use is also extended into digital
signal processing and digital image processing, which utilize digital versions of Fourier analysis
(and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals,
still images, and video signals.
Improper integrals
In applied fields, complex numbers are often used to compute certain real-valued improper
integrals, by means of complex-valued functions. Several methods exist to do this; see methods
of contour integration.
Quantum mechanics
The complex number field is relevant in the mathematical formulations of quantum mechanics,
where complex Hilbert spaces provide the context for one such formulation that is convenient
and perhaps most standard. The original foundation formulas of quantum mechanics – the
Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers.
Relativity
In special and general relativity, some formulas for the metric on spacetime become simpler if
one takes the time variable to be imaginary. (This is no longer standard in classical relativity, but
is used in an essential way in quantum field theory.) Complex numbers are essential to spinors,
which are a generalization of the tensors used in relativity.
Applied mathematics
In differential equations, it is common to first find all complex roots r of the characteristic
equation of a linear differential equation and then attempt to solve the system in terms of base
functions of the form f(t) = ert.
Fluid dynamics
In fluid dynamics, complex functions are used to describe potential flow in two dimensions.
Fractals
Certain fractals are plotted in the complex plane, e.g. the Mandelbrot set and Julia sets.
History
The earliest fleeting reference to square roots of negative numbers perhaps occurred in the work
of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when,
apparently inadvertently, he considered the volume of an impossible frustum of a pyramid,[5]
though negative numbers were not conceived in the Hellenistic world.
Complex numbers became more prominent in the 16th century, when closed formulas for the
roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolo
Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was
only interested in real solutions, sometimes required the manipulation of square roots of negative
numbers. For example, Tartaglia's cubic formula gives the following solution to the equation
x3 − x = 0,
and when the three cube roots of −1 are substituted into this expression the three real roots, 0, 1
and −1, result. Rafael Bombelli was the first to explicitly address these seemingly paradoxical
solutions of cubic equations and developed the rules for complex arithmetic trying to resolve
these issues.
This was doubly unsettling since not even negative numbers were considered to be on firm
ground at the time. The term "imaginary" for these quantities was coined by René Descartes in
1637 and was meant to be derogatory[citation needed] (see imaginary number for a discussion of the
"reality" of complex numbers). A further source of confusion was that the equation
seemed to be capriciously inconsistent with the algebraic
identity
, which is valid for positive real numbers a and b, and which was also
used in complex number calculations with one of a, b positive and the other negative. The
incorrect use of this identity (and the related identity
) in the case when both a and b
are negative even bedeviled Euler. This difficulty eventually led to the convention of using the
special symbol i in place of
to guard against this mistake. Even so Euler considered it
natural to introduce students to complex numbers much earlier than we do today. In his
elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once
and then uses them in a natural way throughout.
In the 18th century complex numbers gained wider use, as it was noticed that formal
manipulation of complex expressions could be used to simplify calculations involving
trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the complicated
identities relating trigonometric functions of an integer multiple of an angle to powers of
trigonometric functions of that angle could be simply reexpressed by the following well-known
formula which bears his name, de Moivre's formula:
De Moivre's Theorem
De Moivre's Theorem is a relatively simple formula for calculating powers of complex numbers.
De Moivre's formula states that for any real number x and any integer n, (cosx + isinx)n =
cos(nx) + isin(nx).
Often abbreviated to:
If n is any integer then
(r cisθ)n = rn cis(nθ)
De Moivres Theorem is an important result in the field of complex numbers, and has practical
applications. It is derived from Euler's formula.
Abraham De Moivre was born in 1667, and was influenced by Sir Isaac Newton. He is best
known for "De Moivre's Theorem", in the field of complex numbers.
De Moivre's Theorem states that for every positive integer, n, the following trigonometric
relationship is true:
(cosx + i.sinx)^n = cos(nx) + i.sin(nx)
where i = √(-1) by convention. (In some literature "j" is used instead of "i", so as not to confuse it
with the symbol for electrical current.)
The result is very useful, especially when trying to sum trigonometric functions.
De Moivre's Theorem Proof
The theorem is derived from Euler's formula
e^(ix) = cosx + i.sinx,
where e = base of natural logarithms, 2.71828...
Raising both sides of the equation to the power of n,
[e^(ix)]^n = (cosx + i.sinx)^n
But since (e^x)^n = e^(x×n) then
e^(i.nx) = (cosx + i.sinx)^n
But by Euler's original formula,
e^(i.nx) = cos(nx) + i.sin(nx), and therefore
(cosx + i.sinx)^n = cos(nx) + i.sin(nx)
De Moivre's Theorem Examples and Applications
When considering De Moivre's Theorem it is important to remember that if two complex
numbers are identical, then the real parts of those numbers must be the same, and the complex
("imaginary") parts of those numbers are the same. i.e.
if
a + i.b = c + i.d, then
a = c, and b = d.
With this in mind, the case for n = 2 can be examined:
(cosx + i.sinx)^2 = cos(2x) + i.sin(2x)
Expanding the left hand side of the equation gives
(cos²x - sin²x) + 2.i.sinx.cosx = cos(2x) + i.sin(2x)
Noting that the real and imaginary parts must equate, then
cos²x - sin²x = cos(2x), and 2.i.sinx.cosx = i.sin(2x)
The formula for cos(2x) and sin(2x) are true, as these are well-known trigonometric identities.
By using the Binomial Theorem, the algebraic expansions for cos(3x), cos(4x), sin(3x) etc can
all be easily obtained. The article "De Moivres Theorem Examples Cos 3x, Sin3x, Cos4x and
Sin4x" describes these in detail.
Trigonometric Series Summation Using De Moivre's Theorem
A further use of De Moivre's Theorem is the summation of trigonometric series. For example, to
find the sum of the series
cos(x) + cos(2x) + cos(3x) + ... + cos(nx)
then the series may be re-written as the real part of the series
(cosx + i.sinx) + (cos2x + i.sin2x) + (cos3x + i.sin3x) + ... + (cos(nx) + i.sin(nx))
This series may in turn be re-written as (using De Moivre's Theorem)
(cosx + i.sinx) + (cosx + i.sinx)^2 + (cosx + i.sinx)^3 + ... (cosx + i.sinx)^n
This series is simply a geometric progression, where
Common ratio (r) = (cosx + i.sinx),
First term (a) = (cosx + i.sinx)
Number of terms = n - 1
So the sum = Real part of (cosx + i.sinx)×((cosx + i.sinx)^n - 1) / ((cosx + i.sinx) - 1)
The rest of this is simply algebraic manipulation, but the steps will be described:
1.
2.
3.
4.
5.
Multiply above and below by the complex conjugate, ((cosx - 1 - i.sinx))
Expand and simply the denominator
Expand and simplify the numerator
Gather real and complex terms
State the result in terms of sinθ and cosθ
De Moivre's Theorem Summary
De Moivre's Theorem is an extension of Euler's formula. It has several uses, such as the
production of formulae for cos(nx) and sin(nx), or the evaluation of the sums of trigonometry.
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