Complex numbers

Definition

A complex number is any number of the form z = a +bi, where a, b \in \mathbb{R} and i is a number such that i^2 = -1.

We say

• a = \Re(z) \in \mathbb{R} is the real part of z
• b = \Im(z) \in \mathbb{R} is the imaginary part of z.

The complex conjugate of z is defined as \overline{z} = \overline{a+bi} = a-bi.

Arithmetic operations

Suppose we have two complex numbers z_1, z_2 where z_1=a_1+b_1i and z_2=a_2+b_2i.

Equality

z_1 = z_2 means that a_1 = a_2 b_1 = b_2

Addition

z_1 + z_2 = (a_1 + a_2) + (b_1 + b_2)i z_1 - z_2 = (a_1 - a_2) + (b_1 - b_2)i

Multiplication

z_1 \cdot z_2 = (a_1+b_1i)\cdot(a_2+b_2i) = (a_1a_2 - b_1b_2) + (a_1b_2+b_1a_2)i

Division

\frac{z_1}{z_2} = \frac{z_1\overline{z_2}}{z_2\overline{z_2}} = \frac{a_1a_2 + b_1b_2}{a_2^2 + b_2^2} + \frac{b_1a_2 - a_1b_2}{a_2^2 + b_2^2}i

You can divide two complex numbers so long as the denominator is non-zero i.e. here z_2\not=0

Algebra

The complex numbers with the above arithmetic operations satisfy:

• commutative rules
  • z_1 + z_2 = z_2 + z_1
  • z_1z_2 = z_2z_1
• associative rules
  • (z_1 + z_2) + z_3=z_1 + (z_2 + z_3)
  • (z_1z_2)z_3=z_1(z_2z_3)
• distributive rule
  • z_1(z_2+z_3) = z_1z_2 + z_1z_3

Geometry

Let z = x+iy \in \mathbb{C}.

The (x,y)-plane is called the complex plane. We call the x-axis the real axis and we call the y-axis the imaginary axis.

The modulus (or absolute value, or length) of z is defined as |z| = \sqrt{z\overline{z}} = \sqrt{x^2+y^2}.

If we represent the complex number z = x+iy as a vector from the origin (0,0) to (x,y), then complex addition, z_1+z_2, corresponds to vector addition (geometrically called the parallelogram rule).

Triangle inequality

|z_1 + z_2|\leq |z_1|+|z_2|.

Polar form

We can write a complex number in polar form using x = r\cos\theta, y = r\sin\theta giving z = x+iy = r(\cos\theta + i\sin\theta). Here \theta is called the argument of z, written \theta = \arg(z), and r=|z| is the modulus of z.

The argument is not unique: if \theta is an argument of z then so too is \theta + 2n\pi for any n\in \mathbb{Z}.

The principal argument of z is that \arg{z} such that -\pi < \arg(z) \leq \pi. It is denoted \mathrm{Arg}(z) and it is unique.

The polar form is useful to multiply, divide, power and root complex numbers: z_1z_2 = r_1r_2[\cos(\theta_1+\theta_2) + i\sin(\theta_1 + \theta_2)] \frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1-\theta_2) + i\sin(\theta_1 - \theta_2)] That is,

• \arg{(z_1z_2)} = \arg{z_1} + \arg{z_2}
• \arg{(z_1/z_2)} = \arg{z_1} - \arg{z_2}
• |z_1z_2|=|z_1||z_2| = r_1r_2
• |z_1/z_2|=|z_1|/|z_2| = r_1/r_2

Powers and Roots

Complex exponential

For a complex varible z = x+iy, the complex exponential function is defined by e^z = \exp(z) = e^{x+iy} = e^x(\cos y + i\sin y).

We also have |e^z| = e^x and \arg(e^z)= y.

Properties:

• e^{z_1}e^{z_2} = e^{z_1+z_2}
• e^{z_1}/e^{z_2} = e^{z_1-z_2}
• Periodicity: e^{z+2\pi i} = e^z

Note thus that saying e^{z_1}=e^{z_2} means that z_1 - z_2 = 2n\pi i for n\in \mathbb{Z}.

The polar form of a complex number, z=r(\cos\theta + i\sin\theta) can be written as z=re^{i\theta}.

Complex logarithm

For a complex varible z = x+iy \not= 0, the complex logarithm is defined as \ln(z) = \log_e|z| + i\arg{z} + 2n\pi i,\quad n\in \mathbb{Z}. Note that this is not a function as it has multiple outputs.

We use \log_e|z| for the real logarithm to distinguish it from the complex log, \ln z. The real logarithm \log_ex has domain x>0, while the complex logarithm \ln z has domain z\not=0.

Properties

• e^{\ln(z)} = z
• \ln(e^z) = z+2n\pi i
• \ln(z_1z_2) = \ln z_1 + \ln z_2
• \ln(z_1/z_2) = \ln z_1 - \ln z_2

The identities are correct up to a term 2n\pi i, n\in \mathbb{Z}.

The principal value of the complex logarithm, \ln z, is defined as \mathrm{Ln}(z) = \log_e|z| + i\mathrm{Arg}(z). This makes the complex log into a single-valued function.