Anonymous
Anonymous asked in Science & MathematicsMathematics · 5 months ago

# De Moivre's Theorem?

Help with this proof please. :)

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• 5 months ago

• 5 months ago

n-th root of r(cos θ + i sin θ) is complex number "z" such that

zⁿ = r(cos θ + i sin θ)

z = ⁿ√(r (cos θ + i sin θ)) = ⁿ√r * (cos θ + i sin θ)^(1/n)

Every complex number has n different n-th roots in the complex plane.

Also, observe that the absolute value (magnitude) ⁿ√r of all the n-th roots is the n-th root of the absolute value "r" of the original number.

At this point, we already may notice that n-th roots (there are "n" of them) are all located on a complex circle with radius equal to absolute value ⁿ√r.

In a complex plane, a complex number on the unit circle is represented as

(cos φ + i sin φ)

where φ is the angle with respect to the real axis

These numbers all have absolute value

cos^2 φ + sin^2 φ = 1

which corresponds to radius of unit circle.

Complex numbers with absolute value (magnitude) "m" are located on a complex circle with radius "m" and are written in the form

m (cos φ + i sin φ)

In our case m = ⁿ√r

z = ⁿ√r * (cos θ + i sin θ)^(1/n)

We need to stop here.

You may already know that, by De Moivre's theorem

(cos θ + i sin θ)ⁿ = cos (nθ) + i sin(nθ)

This applies only for integer powers n. For non-integer powers, de Moivre's formula in this basic form gives inconsistent results. For example, if n=1/2

for θ=0, cos (0/2)+i sin(0/2) = cos (0)+i sin(0) = 1 + i 0 = 1

for θ=2π, cos (2π/2)+i sin(2π/2) = cos (π)+i sin(π) = -1 + i 0 = -1

that is, the result is multiple-valued.

To overcome this limitation so that we can use non-integer power like 1/n, we recall the periodicity of sine and cosine to write

cos θ = cos (θ+2kπ)

sin θ = sin (θ+2kπ)

Where k is an integer

Our n-th root becomes

z = ⁿ√r * (cos ((θ+2kπ)/n) + i sin ((θ+2kπ)/n))

The values of k=0,1,2,…,(n−1) yield distinct values of argument

For fixed n, the argument

φ = (θ+2kπ)/n

have values

θ/n

θ/n+2π/n

θ/n+4π/n

θ/n+6π/n

......

θ/n+2(n-1)kπ/n

Since the angle between any two consecutive complex roots is 2π/n , they are evenly spaced around the circle of radius ⁿ√r

• Vaman
Lv 7
5 months ago

You can write

r cos a+ i r sin a= r exp ( i a +2 m pi). m goes from 1 to infinity.

(r cos a+ i r sin a)^(1/n)= r ^(1/n) exp ( i a/n +2 m/n pi).

Draw a circle of radius 1 unit. Select a. Then n the root will be a/n. Now draw radial lines at angle 2m/n pi for different m values adding a/n.