• Complex Analysis: #1 Complex Numbers

An “imaginary” number is introduced, called i (for imaginary)(sometimes j is also used in physics), which is declared to be a solution of the polynomial equation

x²+ 1 = 0.

The field of complex numbers is denoted by ℂ. We have

ℂ = {x + iy : x, y ∈ ℜ}. 

For z = x + iy, we also write Re(z) to denote the real part of z, namely the real number x. Also Im(z) = y is the imaginary part of z.

Let z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ be complex numbers. Then the addition and multiplication operations are given by

 z₁ + z₂ = (x₁ + x₂) + i(y₁ + y₂)

and

z₁ · z₂ = (x₁·x₂ − y₁·y₂) + i(x₁·y₂ − x₂·y₁).

The complex conjugate is z = x − iy (that is, z = x+i(−y)). Therefore the complex conjugate of z = z . We have |z|² = zz = x² + y². If z ≠ 0 (that is, either x ≠ 0, or y ≠ 0) then zz > 0, and we have

If z ≠ 0 then there are unique real numbers r > 0 and 0 ≤ θ < 2π such that x = r cos θ and y = r sin θ. So let u, v ∈ ℂ be non-zero numbers, and let

u = r(cos θ + i sin θ), 

v = s(cos ψ + i sin ψ). 

Then (remembering the rules for combining trigonometric functions), we see that

u · v = r · s(cos(θ + ψ) + i sin(θ + ψ)). 

If ℂ is identified with ℜ², the 2-dimensional real vector space, then we can identify any complex number z = x +iy. But for any two real numbers x and y, there exists a unique pair of numbers r ≥ 0, 0 ≤ θ < 2π, with x = r cos θ and y = r sin θ. (Obviously it is not quite unique for the number z = 0) So let u = s + it be some other complex number. Then

Thus we see that multiplication of complex numbers looks like an (orientation preserving) orthogonal mapping within ℜ² — combined with a scalar factor r.

More generally, let f : ℜ² → ℜ² be an arbitrary linear mapping, represented by the matrix

 How can this mapping be represented in terms of complex arithmetic? We have

Therefore, using the linearity of f, for z = x + iy we have

Therefore, if f(1) = −if(i), that is, if(1) = f(i), then the mapping is simply complex multiplication. On the other hand, if f(1) = if(i) then we have f(z) = w · z, where w = (f(1) + if(i))/2 ∈ ℂ is some complex number.

for a suitable choice of r and ψ. This is an orientation reversing rotation (again combined with a scalar factor r).

Now let u = a + ib and v = c + id. What is the scalar product <u, v>? It is

Therefore <z, z> = |z|², where |z| = √x² + y² is the absolute value of z = x + iy.