Applications related to complex numbers.

1. Complex numbers and
It’s application

2. HISTORY OF COMPLEX NUMBERS:
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.
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 and multiplication of complex numbers were developed by
the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers
was further developed by the Irish mathematician William Rowan Hamilton.

3. A complex number is a number that can be expressed in the
form a + bi, where a and b are real numbers and i is the
imaginary unit, that satisfies the equation x2 = −1, that is, i2 =
−1. In this expression, a is the real part and b is the imaginary
part of the complex number.

4. Discrete Fourier Transform
The DFT is a ubiquitous algorithm in computer
science, used in image processing, digital
communication, compression and countless other uses
in and around signal processing. It is likely the
most useful and common transformation (linear or
otherwise) in computer science, often being
implemented at the hardware level itself.
Given a sequence of numbers , ,the
DFT is defined as :

5. Complex numbers are
beautiful, because they encode
geometric information
through algebra.

6. Quaternion
In mathematics, the quaternions are a number
system that extends the complex numbers
Quaternions provide a very convenient way of
representing rotations of three-dimensional
space. Even more importantly, when rotations
are represented with quaternions (as opposed
to Euler angles), it becomes much easier to
smoothly interpolate one rotation to another,
which is something computers need to do
repeatedly when animators are working on the
next 3D feature film.

7. Many game programmers have already discovered the wonderful world of quaternions and have
started to use them extensively. Several third-person games, including both TOMB RAIDER titles,
use quaternion rotations to animate all of their camera movements. Every third-person game has
a virtual camera placed at some distance behind or to the side of the player's character. Because
this camera goes through different motions (that is, through arcs of a different lengths) than the
character, camera motion can appear unnatural and too "jerky" for the player to follow the
action. This is one area where quaternions come to rescue.
struct Quat
{
float x;
float y;
float z;
float w;
};
Video Games and Quaternion

8. "The shortest route between
two truths in the real domain
passes through the complex
domain."

9. A fractal is a natural phenomenon or a
mathematical set that exhibits a repeating
pattern that displays at every scale.
Fractal
With computers, we can generate
beautiful art from complex numbers.
These designs are called fractals.
Fractals are produced using an iteration
Common fractals are based on the Julia Set and the
Mandelbrot Set.

10. The Julia Set equation is: Zn+1 = (Zn)2
+ c
For the Julia Set, the value of c remains constant and the
value of Zn changes
The Mandelbrot Set
The Mandelbrot is the same as the Julia Set, but the
value of c is allowed to change.

11. Analytic combinatorics
In mathematics, analytic combinatorics is
one of the many techniques of counting
combinatorial objects. It uses the internal
structure of the objects to derive formulas
for their generating functions and then
complex analysis techniques to get
asymptotics.
It is used for analysis of algorithms.

12. The fast multipole method has been
called one of the ten most significant
algorithms in scientific computation
discovered in the 20th century.
Fast multipole method
The fast multipole method (FMM) is a
numerical technique that was
developed to speed up the calculation
of long-ranged forces in the n-body
problem. It does this by expanding the
system Green's function using a
multipole expansion, which allows one
to group sources that lie close together
and treat them as if they are a single
source.

13. N-Body problems inevitably come up when
doing any most any kind of physical
simulation work, particularly when particles
are involved.

14. The end