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complex analysis applications- hasibul ahsan

Complex number is defined on argon plane. The abscissa is the real number line and the ordinate is the imaginary number (squreroot(-1)) line. So a point on the argon plane can be represented as a+ib where a is the number of units that represent the abscissa and b is the number of imaginary units i. Argoon plane or complex plane looks like this :
complex plane
The use of imaginary number is pervasive in physics and engineering. It simplifies a lot of calculation. Electricity , for example , can ba modeled with imaginary quantity. In quantum field theory all the integrations are computed using this imaginary number. The algebra of complex numbers is different. It is more like vector algebra. For example multiplication of two complex numbers would form a complex number with different amplitude and different angle.

complex multiply
A typical quantum field is repsented by this formula

complex multiply
An important theorem in complex analysis is the Demoiver theorem which relates cosine and sine with the imaginary quantity i.
Demoiver theorem
Nth root of unity is expressed using imaginary number:

Demoiver theorem
So all solutions when raised to power n give unity or 1.

"Reality is like a limit , we approach to it but can never touch it or reach it"

Cauchy integral formula

It is one of most remarkable theorems in complex analysis. If a function is analytic in domain R then cauchy's integral formula tells :

Cauchy's theorem
That is if you know the values of the function f(z) along a contour C you can determine its value in anywhere inside the contour. And you can determine its derivative also.
Cauchy's theorem
This brings us to contour integration.

Contour integration

Contour integal is defined along a contour or closed loop in complex plane. That is the range of the values of the function f(z) falls on the contour curve. An example can make it clear

Cauchy's theorem

Contour integration

In evaluating those integral we have used estimation lemma which states as follows:
In mathematics the estimation lemma, also known as the ML inequality, gives an upper bound for a contour integral. If f is a complex-valued, continuous function on the contour Γ and if its absolute value |f(z)| is bounded by a constant M for all z on Γ, then

Contour estimation lemma

When calculating contour integral a branch cut might fall inside the contour. In that case the contour needs to be distorted to evade the branch cut. A A branch cut is a curve (with ends possibly open, closed, or half-open) in the complex plane across which an analytic multivalued function is discontinuous. For convenience, branch cuts are often taken as lines or line segments. So in case of branch cuts the integral becomes troublesome. A roundabout way is applied

Cauchy's theorem and brunch cut
The integation done above is a kind of improper integral. Improper integral is the integral which gives infinite value or beomes divergent at the limit point. So a dummy variable η is used here. It is infinitesimal quantity which tends towards zero. More can be said about improper integral , which at the moment best left. The most notable example of this kind of improper integral can be seen in Feynman's propagator's derivation. Another example of improper integral is the integral of 1/x :

improper integral
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Complex analysis for mathematics and engineering continues

Extended Cauchy's theorem

Cauchy's theorem

Some theorems proved by Euler

Euler's equations
The fourth equation is the famous equation related to graph theory. E represents the number of edges , F represents the number of faces and V represents the number of vertices. When this is the case then the equation always holds true.

Some more complex formulas

complex equations

matrix and mathematics

Couchy schwarz inequality

Couchy  schwarz inequality

Maclaurin series

Maclaurin series is the special case of Taylor's series where we evaluate the function and its derivative at the origin 0. Here is the example:

Maclaurin series
I was once thinking how calculator calculate the values of trigonometric functions like sines, cosines and tangent . It is possible that algorithm is developed to evaluate the Maclaurin series. This way it is easy to calculate those.

Gamma Function

Gamma function is a special type function . It can be represented with complex number as follows:
Gamma function

Simply connected domain

A picture is worth a thousand words.
Gamma function

Real Analysis

Real Analysis of sequence

Reference materials:

Law of thermodynamics
A briefer history of time by S. Hawking
A brief history of time by S. Hawking
Quantum mechanics
Grand Design by Stephen Hawking
perihelion of mercury by Feynman
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