1
Complex Analysis
x2 + 3 = 0
x 2 − 10x + 40 = 0
Real solutions are not feasible!!
Definition : An ordered pair (x, y) ≡ z ∈ ℂ such that Re(z) ≡ x ∈ ℝ & Im(z) ≡ y ∈ ℝ
If z1 ≡ (x1, y1) & z2 ≡ (x2, y2) then
Equality of two complex numbers: z1 = z2
Imaginary unit: i=(0,1) such that
⟹
z1 + z2
z1 × z2
≡
≡
(x1 + x2, y1 + y2)
(x1x2 − y1y2, x1y2 + x2 y1)
Re(z1) = Re(z2) & Im(z1) = Im(z2)
i2 = − 1
Complex plane: Argand diagram
∈ℂ
∈ℂ
y
z ≡ (x, y) ≡ x + iy
Im axis
Re axis
Complex numbers are an extension of real numbers such that Im(z)=0
Unlike real numbers complex numbers can not be ordered as in a sequence
∀p ∈ ℝ & p ≥ 0 ⟹ p 2 = p . p ≥ 0
i > 0 ⟹ i 2 = i . i = − 1 ≱ 0 contradiction ×
x
1
Complex number: z = x + i y = (x,y)
| z |2 = z . z * = (x + iy) . (x − iy)
Conjugate: z* = x - i y
= (x, y) . (x, − y) = (x 2 + y 2,0) ≡ x 2 + y 2 ∈ ℝ
x2 + y2
Modulus: | z | =
Polar form: x = r cos θ
y
& y = r sin θ
Im axis
z = (x,y) = x + i y = r cos θ + i r sin θ = r[cos θ + i sin θ]
θ
Re axis
x
e −iθ(cos θ + i sin θ)
0
f(θ) = constant
f(0) = 1 ⟹ f(θ) = 1
□
e iθ = cos θ + i sin θ
Euler’s formula
Let
f(θ)
f′(θ)
z = re iθ = | z | e iArg(z)
r = |z| =
r
2
x +y
2
&
z = (0,0) ⟹
−1
Arg(z) = θ = tan
θ
is undefined
y
x
=
=
hence
since
−π ≤θ ≤π
principal values
1
z1 = r1(cos θ1 + i sin θ1) ≡ (r1 cos θ1, r1 sin θ1)
z2 = r2(cos θ2 + i sin θ2) ≡ (r2 cos θ2, r2 sin θ2)
z1 . z2 = r1r2[(cos θ1 cos θ2 − sin θ1 sin θ2) + i (sin θ1 cos θ2 + cos θ1 sin θ2)]
= r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)]
Modulus:
| z1 . z2 | = | z1 | . | z2 | = r1r2
Argument:
arg(z1 . z2) = arg(z1) + arg(z2) = θ1 + θ2
If
z1 = z2 = z = re iθ
| z2 | = r2
arg(z 2) = 2arg(z) = 2θ
De Moivre’s formula
z n = r n(cos θ + i sin θ)n = r n[cos nθ + i sin nθ]