De-Moivre’s Theorem and Euler Formulas

If you are looking for help on raising complex numbers to power have a look at De-Moivre’s Theorem and Euler Formulas provided. Use the De-Moivre’s Theorem and Euler Formulae List existing to make your calculations much more simple and fast. Try to memorize the Simple Formulas prevailing and understand the concepts easily.

List of De-Moivre’s Theorem and Euler Formulas

General De-Moivre’s Theorem and Euler Formulas are stated below and you can make the most out of them. Learn the concept easily and overcome the hectic task of calculations by referring to the formulae over here. Quickly grasp them and do it the right way while solving your problems.

1. De- Moiver’s Theorem:

It states that if n is rational number (positive, negative or zero) then
(cos θ + i sin θ)n = cos nθ + i sin nθ &
(cos θ + i sin θ)-n = cos nθ – i sin nθ ; n ∈ Q

2. Euler’s Formula

e = cos θ + i sin θ and e-iθ = cos θ – i sin θ
e + e-iθ = 2 cos θ and e – e-iθ = 2i sin θ

3. nth roots of complex number (z1/n)

= r1/n $$\left[\cos \left(\frac{2 m \pi+\theta}{n}\right)+i \sin \left(\frac{2 m \pi+\theta}{n}\right)\right]$$ where m = 0, 1, 2, …….., (n – 1)

• Sum of all roots of z1/n is always equal to zero
• Product of all roots of z1/n = (-1)n-1 z

4. Cube root of unity

1 + ω + ω2 = 0, ω3 = 1 where ω = – $$\frac{1}{2}+\frac{i \sqrt{3}}{2}$$

5. Continued product of the roots

• Continued product of roots of z1/n = $$\left\{\begin{array}{l}\mathrm{z}, \text { if } \mathrm{n} \text { is odd } \\-\mathrm{z}, \text { if } \mathrm{n} \text { is even }\end{array}\right.$$
• Continued product of values of zm/n = $$\left\{\begin{array}{c}z^{m}, \text { if } n \text { is odd } \\(-z)^{m}, \text { if } n \text { is even }\end{array}\right.$$

6. The sum of pth powers of nth roots of unity

$$$$=\left\{\begin{array}{l}n \text { when pis a multiple of } n \\0 \text { when } p \text { is not a multiple of } n\end{array}\right.$$$$

7. Some important results

If z = cos θ + i sin θ

• z + $$\frac{1}{z}$$ = 2 cos θ
• z – $$\frac{1}{z}$$ = 2 i sin θ
• zn + $$\frac{1}{z^{n}}$$ =2 cos nθ, zn – $$\frac{1}{z^{n}}$$ = 2i sin nθ
• If x = cos α + i sin α , y = cos β + i sin β, z = cos γ + i sin γ and given, x + y + z = 0, then
• $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$ = 0
• yz + zx + xy = 0
• x2 + y2 + z2 = 0
• x3 + y3 + z3 = 3xyz