# 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^{iθ} = cos θ + i sin θ and e^{-iθ} = cos θ – i sin θ

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

**3. n ^{th} roots of complex number (z^{1/n})**

= r^{1/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 z
^{1/n}is always equal to zero - Product of all roots of z
^{1/n}= (-1)^{n-1}z

**4. Cube root of unity**

1 + ω + ω^{2} = 0, ω^{3} = 1 where ω = – \(\frac{1}{2}+\frac{i \sqrt{3}}{2}\)

**5. Continued product of the roots**

- Continued product of roots of z
^{1/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 z
^{m/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 p ^{th} powers of n^{th} roots of unity**

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

**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
- z
^{n}+ \(\frac{1}{z^{n}}\) =2 cos nθ, z^{n}– \(\frac{1}{z^{n}}\) = 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
- x
^{2}+ y^{2}+ z^{2}= 0 - x
^{3}+ y^{3}+ z^{3}= 3xyz