AN ALTERNATE PROOF OF THE DE MOIVRE'S THEOREM

Abstract

 Transformation is a method for solving the system, where we face difficult. As the well known 
transformations like Fourier transformation, Laplace transformation, Z-trans formations helps us to solve 
different equations like differential algebraic problems. In   this paper I propose a new transformation called C
transformation, from the set of complex numbers onto the set of all  2 2× matrices of the form a ib
ib a
 
   
 
where a, b are real numbers. I defined a new set M, which is nothing but the set of all    2 2× matrices of the 
form 
a ib
ib a
 
   
 where a, b are real numbers. The C-transformation is a transformation   from the set of 
complex numbers   onto M. I further show that   the C-transformation is isomorphism from C onto M. The 
existence and uniqueness of such transformation is proved. By using this transformation, I try to give the 
geometrical    interpretation for the matrix of the form 
a ib
ib a
 
   
  which is nothing but a point in the argand 
plane. I gave an alternate proof for the well known Demoiver’s theorem. I further try to solve a conformal 
mapping from z-plane  to w-plane using this C-transformation provided the angle preserving , ie θ= φ( the 
angle  between the curves in z-plane is equal to the angle between the curves in w-plane ) with an example 
w=f(z)= 2z at the point z=1+i.