Moivre’s Law . In Mathematics and more specifically Algebra and Mathematical Analysis expression that expresses the power (initially of natural and then real exponents) of normal complex numbers in their polar representation, due to Abraham de Moivre ( 1667 – 1754 ) in 1697 .
Its importance is to abound in the relationship between trigonometry and complex potentiation and that the result, when it can be extended to real and complex powers, allows the calculable definition of any base power and complex exponents.
Summary
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- 1 Definition
- 2 Obtaining
- 3 Importance
- 1 Calculation of nth roots of a complex number
- 4 See also
- 5 Sources
Definition
Let be a purely imaginary unitary complex number cis x , where x is any real representing an angle in radians and n any natural; to the expression:
- (cis x) n= cis (nx) = cos (nx) + i sin (nx)
it is known as De Moivre’s Law or Formula .
Obtaining
Starting from the well-known Euler’s Law :
- and ix= cis x
If both members are raised to any real n (this obviously includes the natural ones) we obtain:
(e xi ) n = (cis x) n
and developing the left member of the equality is:
e nxi = cis (nx) = cos (nx) + i sin (nx)
To finally find De Moivre’s formula:
(cis x) n = cis (nx) = cos (nx) + i sin (nx)
Importance
On the one hand, De Moivre’s formula allows associating multiple angle trigonometry with the power of complexes and extending this same idea to powers of real and complex numbers, which together with the idea that each complex number expressed in polar notation is multiple due at full rotations (360 0 ) of each argument, this multiplicity radiates to complex power, exponential, and logarithm operations.
Calculation of nth roots of a complex number
This law allows us to clarify the fact that any complex number z = a cis b has n nth roots , but it also expresses its geometric meaning: each root is a vertex of a regular n-gon inscribed in a circle with center in the origin and radius , located at the angles .
For example, when calculating the sixth root of 64 cis 30 o , you must first rewrite the number so that the angles matching 30 o are included in the argument . Now the expression remains:
- (64 cis (30 or+360 or k)) 1/6 ; where k is any integer .
- = 64 1/6cis (30 o / 6 +360 o k / 6)
- = 2 cis (5 or+60 or k)
and giving k the values 0, 1, 2 …, 5, the sixth roots are obtained:
- 2 cis 5 o, 2 cis 65 o , 2 cis 125 o , 2 cis 185 o , 2 cis 245 o , 2 cis 305 o .
whose geometric representations would be:
