[THEOREMS] Polar form

Modulus laws

For $z,z_1,z_2\in\mathbb C$ with $z_2\ne0$ ,

z\overline z=|z|^2,\qquad |z_1z_2|=|z_1||z_2|,\qquad \left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|},

|\operatorname{Re}(z)|\le|z|, \qquad |\operatorname{Im}(z)|\le|z|, \qquad |z_1+z_2|\le|z_1|+|z_2|,

|z_1-z_2|\ge\bigl||z_1|-|z_2|\bigr|, \qquad \left|\sum_{j=1}^N z_j\right| \le\sum_{j=1}^N|z_j|\quad(N\ge1).

The inequality for $z_1+z_2$ is the triangle inequality in the Argand plane; the finite-sum version follows by applying it repeatedly. The reverse triangle inequality follows from

|z_1|=|(z_1-z_2)+z_2|\le |z_1-z_2|+|z_2|

and the same argument with $z_1$ and $z_2$ interchanged.

[1]:Complex modulus

De Moivre's formula

For $\theta\in\mathbb R$ and $n\in\mathbb Z$ ,

\left(\cos\theta+i\sin\theta\right)^n =\cos(n\theta)+i\sin(n\theta).

More generally, if $z=re^{i\theta}$ with $r>0$ , then

z^n=r^ne^{in\theta}.

For positive integers, use induction. The case $n=1$ is immediate and if the formula holds at $n=k$ , multiplying by $re^{i\theta}$ gives

z^{k+1}=r^ke^{ik\theta}re^{i\theta}=r^{k+1}e^{i(k+1)\theta}.

The zero case is $z^0=1$ and the negative-integer case follows from the multiplicative inverse. The condition $r>0$ is essential for negative powers, because $0$ has no multiplicative inverse.

Find the nth roots

If $z_0\ne0$ , $n$ is a positive integer and $z_0=r_0e^{i\theta_0}$ , its distinct $n$ th roots are

z_k=\sqrt[n]{r_0}\exp\left(i\frac{\theta_0+2\pi k}{n}\right), \qquad k=0,1,\ldots,n-1.

They lie on a circle of radius $\sqrt[n]{r_0}$ centred at the origin and are equally spaced by an angle of $2\pi/n$ . A non-zero complex number has exactly $n$ distinct $n$ th roots.

To derive the formula, write $z=\rho e^{i\phi}$ . If $z^n=z_0$ , equality of moduli gives $\rho^n=r_0$ , while equality of arguments gives $n\phi=\theta_0+2\pi k$ . Taking $k=0,\ldots,n-1$ produces all distinct possibilities: two listed angles differ by less than $2\pi$ , so they cannot describe the same point. Replacing $k$ by $k+n$ adds $2\pi$ to the root's angle, so any other integer $k$ repeats one of them. If $z_0=0$ , the only $n$ th root is $z=0$ .

Theorems from Polar form

Modulus laws

De Moivre's formula

Find the nth roots

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