BodhiAI

Nov. 12, 2019
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 I

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)