Analysis of Real and Complex Numbers

Analysis
The study of analysis means studying infite.

Sequence
Sequence are function whose domain is natural number and range is real number or complex number depending on the context.

Real Number

History and Challenges of Complex Number
Origin of Complex Numbers does present a challenge to anybody. Most of us are introduced to the complex numbers in our grade 8 when we learn to solve quadratic equation and we encounter sqrt(-ve number). I am currently taking complex analysis so I am fortunate to read some books and broswe through websites, so here is an easy way to understand these mysterious, imaginary numbers in a concrete way. Historically five people whose name comes up have an interesting story. These people are Del Ferro,Fior,Tartaglia,Cardano, Ferrari and Bombelli.
Del Ferro was the first person who is known to have come up with the solution of the cubic equation, since he didn't published much infact none of his publication has survived. The legend is he revealed his work to one of his pupil called Fior. However he revealed only one part of the solution. Not realizing this Fior become overconfident in his mathematical ability and met Tartaglia in a mathematical duel. Tartaglia solved all the 30 problems posed by Fior and established himself a force in mathematics from being a simple self taught teacher. History doesn't pain Tartaglia in a very good light. He is said to have not published his results so that he could won more mathematics competition. Then arrives Cardano a person who was an illegitimate, genius mathematician. Whose story is from rags to riches and then dying in ignomity. He was an Italian who tried several times to gain entry into influential mathematical circle but couldn't gain it until he proved his mettle by writing books and excelling in treating his patients. Once he arrived in the mathematical scence he was a force to reckon with. He persuaded Tartaglia to reveal his formula for cubic equation but Tartaglia entered into a secret pact with him not to reaveal it to other. Cardano had a terrific protege whose name was Ferrari and they discovered that its not Tartaglia who first discovered the formula for solving cubic equation but it was Del Ferro. So Del Ferro thought his promise to Tartaglia was not binding to him anymore and he published the result in his celebrated text Art magna. Tartaglia was infuriated and challenged him for mathematical duel. After much deleberations Tartaglia finally agreed to take on Ferrari, Cardano's protege. Much to his chagrin he finds him very much familiar with the nuances of mathematics and Tartaglia fleds the very same day. Thus Ferrari comes out of the shadow of Cardano. He goes on to find the general solution of fourh power equation and lived on a happy life unlike Cardano. Who suffers ignominity because of his son's act to poison his wife and his name in that conspiracy. His son was decapitated. The other son is equally incompetent and that tears apart Cardano. Finally comes Bombelli who after looking at the formula sees what all had previously overlooked and dismissed as unimportant and that is Bombelli's wild thought.

Well today I also got to know that there are numbers called Quaternions which are superset of Complex Numbers. As complex numbers are defined for 2 dimensions. Quaternions are defined for four dimensions.

Studying complex analysis is very much similar to any other analysis class and infact some of the theorems like Bolzanno Wierstrauss, Heine Borel do pop up in complex analysis.

Application of complex Numbers
There are lot of places where complex numbers are used. One is while studying Boundary Value Problems in Heat flow. For a simple disc if we keep it heated at the boundary the temperature at the centre is the average value at the boundary. For example if 3/4 of the boundary of the disc is at 0 degree and other quarter is at 100 degree then the tempeature at the centre of the disc is 25 degree centigrade.
Complex numbers also find application in things like finding the nth root fot eh equation. One thing where complex numbers rule is the Fundamental theorem of Algebra.
Here is a list of common place where we can use complex numbers.

Complex Number

Advantage of Complex numbers

Riemann's Contribution
Zeta Function, Riemann Sphere
Riemann is known for extending the concept of Zeta function from real to complex number. Zeta function was discovered by Euler and it gives a relation between prime number and the infinite series. For real numbers it looks like Zeta(s) = Summation (1/(n^s)) where s is any real number > = 1. Note for s =<>

Cauchy's Contribution
Cauchy and Riemann have affected the development of Complex number so much. Cauchy
Cauchy's Theorem for Analiticity
Cauchy's theorem for Analiticity says that

Laplace Equation
Laplace equation is a fundamental eq in many fields of physics and engineering and it gives a test for very important harmonic function. If we have a function U(x,y) then its harmonic if we can find
Uxx+Uyy = 0. Note here Uxx means second partial with respect to x and Uyy is second partial with respect to y

Topology to know for Complex Numbers
The first thing is we will define Neighbor in set notation and from there all other definitions of Interior pt, Exterior pt, Boundary pt, Open Set, Close set etc will follow.

Neighborhood Its very easy to define neighbor in terms of delta. N(a,delta) ={x abs(x-a) <>Interior Point The interior point is all points
Open Set Usually open set is denoted by letter G. It includes all the points in the domain except the boundary points. In set notation it will be that we can find a delta Neighborhood > 0 such that all the points in the neighbor are subset of the set.
Closed Set Closed set enclose the boundary also. Other definition of close set which maks more sense is that it contains all the limit points inside the set. So if we have a converging sequence like {1/n, n belongs to N}, we can see that this sequence is bounded between 0 and 1. However the set is not closed because it doesn't contains zero the limit point of the sequence 1/n. (If you may be wondering about the series 1/n, then yes its not convering, 1/n is a diverging series. What it means is if you do sum like 1+1/2+1/3+1/4+1/5... that sum will goto infinity when n tends to infinity) but as far as sequence 1/n concerns its converging.

Connection
Simply Connected
A set is simply connected if you can draw a polyonal line from one pt to another without going outside the domain ie each point on the line is inside the domain.

Multiply Connected
If there are holes in the domain set then there are multiple ways you can go from one pt to another and this is called multiply connected.

Compact Set
Set which are bounded and closed are called Compact set. Thats a Heine Borel theorem. This gives an easy way to say that all open sets are not compact. However if some one ask you to define Compact set. Then you got to know what is a Cover. A cover is basically a union of sets. So a cover looks like {{}.{},{},{} ...} and its CoverC = {UAi, where Ai is elesment of CoverC, U stands for union}. Cover is basically an open set. Coming back to our original thought a set is said to be Compact set if every cover has a finite subcover. So what is a finite subcover ? A finite subcover is something which covers the set
To prove why the interval (0,1) is not compact
By Heine Borel theorem we can say that since its not close its not compact. Think of some subcovers for (0,1)
for ex (-1,.5) U(.5,1,5) is one subcover. Now the definition says that every subcover has to have a finite subcover.
We can actually come up with a cover for example
{1/n,n}
Why set [0,1] is compact. The first thing we note that its close and second thing we note that its bounded so its compact.

Now we list two famous theorem of intervals. Note both these theorems tell that if we have nested interval or Rectangle or we can extend to any other geometrical figure than we will have a unique pt which will be common to all the other figures.

Nested Interval Theorem
This theorem says that if we have a sequence of interval i1,i2,i3...
1. Each interval i(n+1) nested inside i(n)
2. As n-> infinity, length of interval -> 0
Then there will be a only one pt in the interval and it will be common to all the interval.

Nested Rectangle Theorem
This theorem says that if we have a sequence of nested rectangles r1, r2...
1. Each rectangle r(n+1) nested inside r(n)
2. As n-> infinity, length of diagonal -> 0
Then there will be a only one pt in the rectangle and it will be common to all the rectangles.


Hine Borel's Theorem
Hine Borel's Theorem links the infinite pts in complex domain to something manageable. The statement of this theorem is "Every close and bounded set is compact". Compactness is not easy to define but using Heine Borel's theorem it becomes so simple. Note Heine and Borel are two persons not one. Compactness is defined in terms of cover. Where cover is a union of sets.

Jordon Curve
Well Jordon curve also has an interesting history. The theorem is so obvious yet its general proof is not so easy. Infact at the time of its statement there were not enough mathematical tools to prove it. No doubt the proof given by the Jordon turned out to be wrong ! Later its proof was given,
The theorem says that a "simple close curve" divides the plane into two classes. What we mean by simple close curve is "imagine you have a rubber band, now you can strech it in any number of directions and you still get a polygon curve, like ellipse, pentagon, hexagon etc. However the edges shouldn't cross for example you cannot make a shape 8 where the edges are overlapping.
Let us call them class A (pts insides the close curve) and class B(points outside the close curve). Jordon's theorem says that any point in class A can be connected with the points by a polygonal line without needing to cross the boundary of that class.
Convergence

Limit, Supremum, Infimum
To have a limit means all the point in the neighbourhood converge to the limit. That is where delta epsilon definition comes in
Continuity of Complex Number
We define continuity of complex number in the context
Analytic function
What are analytic functions ? Analytic functions are differentiable in every pt of their open domain (G) and we say its analytic at a pt then it means it is analytic in some neighborhood of that pt.

Integration in Complex Plane
Integration in complex plane is bit different. For example integral using Cauchy's integral we can integrate