Complex numbers · Definition
De Moivre's formula
Before you start you need to know
It turns raising a complex number to an integer power into an operation on modulus and angle, with no binomial expansion.
Definition
De Moivre's formula gives the integer power \(n\) of a complex number \(z\) of modulus \(r\) and argument \(\theta\): it raises the modulus to the \(n\)-th power and multiplies the argument by \(n\).
If \(z=r(\cos\theta+i\sin\theta)\), the product rule in polar form says that multiplying adds angles and multiplies moduli. Applying it \(n\) times gives \(z^n=r^n(\cos n\theta+i\sin n\theta)\). It is a direct induction on \(n\), and the formula still holds for \(n=0\) and \(n<0\) because \(z^{-1}\) has modulus \(1/r\) and argument \(-\theta\).
For example, for \((1+i)^8\) write \(1+i=\sqrt2(\cos\frac\pi4+i\sin\frac\pi4)\) to get \((\sqrt2)^8(\cos2\pi+i\sin2\pi)=16\), far shorter than multiplying eight times. Geometrically, a power stretches the radius and multiplies the angle: a spiral when \(r\neq1\).
When \(r=1\) we get \((\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta\); comparing real and imaginary parts with the binomial theorem yields identities such as \(\cos3\theta=4\cos^3\theta-3\cos\theta\). This is one reason polar form is worth learning.
The formula is for integer exponents. With fractional exponents, such as \(\frac1n\), the argument \(\theta\) and its equivalents \(\theta+2\pi k\) give different results; that is the case of \(n\)-th roots, the next concept.
Think about it
What is \(\left(\frac{1}{2}+\frac{\sqrt3}{2}i\right)^{6}\)?
It is \(\cos\frac\pi3+i\sin\frac\pi3\) to the 6th power, i.e. \(\cos2\pi+i\sin2\pi=1\).
Why does the formula not depend on the chosen argument when \(n\) is an integer?
Because replacing \(\theta\) by \(\theta+2\pi\) changes \(n\theta\) by \(2n\pi\), a multiple of \(2\pi\), which sine and cosine ignore.
Exercises 10
- Compute \((1+i)^8\).
- Compute \(\left(\cos\frac\pi6+i\sin\frac\pi6\right)^6\).
- Use the formula to find \((1-i)^4\).
- What is the modulus of \((\sqrt3+i)^5\)?
- Compute \((1+i\sqrt3)^3\).
- For \(n=-1\), \(\left(\cos\theta+i\sin\theta\right)^{-1}\) equals:
- Expanding \((\cos\theta+i\sin\theta)^3\) and equating real parts, what expression results for \(\cos3\theta\)?
- Compute \((\sqrt3-i)^6\).
- If \(n\) is an integer and you replace \(\theta\) by \(\theta+2\pi\) in \(r^n(\cos n\theta+i\sin n\theta)\), the value:
- What is the smallest positive integer \(n\) such that \((1+i)^n\) is a positive real number?
What comes next
References
- Elias M. Stein, Rami Shakarchi, Complex Analysis, 1st edition, Princeton University Press, 2003. Cap. 1, §1.1
- James Ward Brown, Ruel V. Churchill, Complex Variables and Applications, 9th edition, McGraw-Hill, 2014. §7
- Lars V. Ahlfors, Complex Analysis, 3rd edition, McGraw-Hill, 1979. Cap. 1, §2.1