26 Jul 2025

Complex Numbers, Euler's Formula, De Moivre's Theorem, and Roots

Argand-plane geometry, polar form, complex multiplication, integer powers, and all roots of a complex number.

bsc semester-ii mathematical-physics complex-analysis complex-numbers

A complex number is

\[z=x+iy, \qquad i^2=-1,\]

and is represented by the point $(x,y)$ in the Argand plane. Its conjugate and modulus are

\[\bar z=x-iy, \qquad \lvert z\rvert=\sqrt{z\bar z}=\sqrt{x^2+y^2}.\]

For $z\ne0$, let $\theta$ be an argument of $z$. Then

\[x=r\cos\theta,\qquad y=r\sin\theta,\qquad r=\lvert z\rvert,\]

so

\[\boxed{z=r(\cos\theta+i\sin\theta)}.\]

Because $\theta$ and $\theta+2\pi k$ represent the same point, the argument is multivalued. A selected principal argument is commonly restricted to one interval of length $2\pi$.

Euler’s formula

Let

\[w(\theta)=\cos\theta+i\sin\theta.\]

Differentiation gives

\[\frac{dw}{d\theta} =-\sin\theta+i\cos\theta =i(\cos\theta+i\sin\theta) =iw,\]

with $w(0)=1$. The exponential $e^{i\theta}$ is the solution of the same equation $w’=iw$ with the same initial value. Hence

\[\boxed{e^{i\theta}=\cos\theta+i\sin\theta}.\]

Therefore the polar form is

\[\boxed{z=re^{i\theta}}.\]

Multiplication now has a direct geometrical meaning:

\[z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}.\]

The moduli multiply and the arguments add.

De Moivre’s theorem

For an integer $n$,

\[(\cos\theta+i\sin\theta)^n =(e^{i\theta})^n=e^{in\theta}.\]

Using Euler’s formula again,

\[\boxed{ (\cos\theta+i\sin\theta)^n =\cos(n\theta)+i\sin(n\theta)}.\]

For negative $n$, the result follows by taking reciprocals, since $\lvert e^{i\theta}\rvert=1$.

Roots of a complex number

Let

\[z^n=Re^{i\Theta}, \qquad R>0.\]

Write $z=re^{i\theta}$. Then

\[r^ne^{in\theta}=Re^{i(\Theta+2\pi k)}.\]

Equality of moduli and arguments gives

\[r=R^{1/n}, \qquad \theta=\frac{\Theta+2\pi k}{n}.\]

Only $k=0,1,\ldots,n-1$ give distinct roots, because increasing $k$ by $n$ adds $2\pi$ to $\theta$. Thus

\[\boxed{ z_k=R^{1/n} \exp\!\left[i\frac{\Theta+2\pi k}{n}\right], \quad k=0,\ldots,n-1}.\]

The roots \(z_k=R^{1/5}e^{i(\Theta+2\pi k)/5}\) lie at equal angular intervals \(2\pi/5\). Their polygon is equation-generated from the root formula.

For example, the cube roots of $8=8e^{i2\pi k}$ are

\[z_k=2e^{i2\pi k/3}, \qquad k=0,1,2,\]

or

\[\boxed{2,\quad -1+i\sqrt3,\quad -1-i\sqrt3}.\]

Cubing each value gives $8$, as verified in the Unit III Maxima worksheet.