Complex Analysis Notes ,Advanced mathematics part 2, Handwritten Notes

 

circle

disk and neighborhood

limits of real functions

graphic representation of limits

criteria for non-existance of limit

limit don't exist

limit exist and unique

real and imaginary part of limit

compute limit

compute limit

Properties of complex limits

theorems of limits

limits theorems

continuity

example of continuity

differenbility

if f(z) is differentiable then f(x) is continuous

analytic function at a point ,analytic in a domain

entire function, singular points

conjugate of function is not analytic,cauchy-Riemann equations or CR's equations

criteria for non-analytically

importants point about analyticty

sufficient condition for differentability

function is analytic at origin

harmonic function/potential function,harmonic conjugate function

obtain conjugate and the original function

example of harmonic function

cr equation in polar form

analytical function with constant modulus is constant

function is analytic and its derivative is zero ,is this constant?

analytic test

complex exponential function

derivative,conjugate,argument of complex function

modulus ,algebraic properties of complex exponential function,periodic function

exponential function is periodic

theorems related to periodicity of complex exponential function

complex logarithmic function

algebraic properties and principal value of complex logarithmic function

logarithmic function is inverse of exponential function

analyticity of principal branch of complex logarithmic function

complex powers ,principal value of complex powers

complex trigonometric functions

periodicity 

Module theory Notes, Advanced mathematics, Abstract Algebra notes, Handwritten notes

Module definition

 

Module Example

Module Example

Z_module 

Z_module and the difference between vector space and module

sub-module criteria

example of sub-module

examples of sub-module

sub-module examples solution

sub-module examples solution

sub-module example solution

is R^n be sub-module?

Rn be sub-module

module homomorphism

every ring homomorphism is module homomorphism but the converse is not true.

a map is an R-module homomorphism

an abelian group is a ring homomorphism

elements of ring homomorphism are module homomorphism

ring homomorphism is group

ring homomorphism is abelian

endomorphism, quotient module, R-module is homomorphism with kernel

Quotient group under the action of ring elements

projection map is a module homomorphism

sub-module, the sum of the module,1st isomorphism theorem of the module

2nd,3rd,4th isomorphism theorems of module

 proposition

pi is surjective

cyclic module

finitely generated module

R module is finitely generated 

direct product, free module

free module example, direct product is R module 

Module fundamentals