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BodhiAI
Nov. 12, 2019

De Moivre's theorem:

De Moivers Theorem

De-Moivre's Theorem is  relatively simple formula for calculating powers of complex numbers. There are two possible cases. 

Case I If n is an integer

If n is any Integer, then

Case II If n is a Fraction 

If n is any Rational Number i.e. such that 

a)

b) p, q

c) q > 0, and

d) p and q have no common factors, then

z =

has 'q' distinct values one of them is

Important

Let z = be any complex number, then by De Moivre's Theorem has n distinct values of z or n roots of z and one of them is

Roots of Any Complex Number 

Let be any complex number having modulus r and argument , such that

z = r

is the polar equivalent form of z

Now, if m and n be any integers, such that n is positive and m and n have no common factors, then   

{De-Moivre's Theorem for integer}

Now, will posses n distinct values / roots of and they are

where, k = 0, 1, 2, 3, ........., (n–1)