Engng Math 145 Complex numbers

Rectangular form

z = x +iy

Complex conjugate

z̄ = x - iy

Rectangular form +, -, *, /

Conjugate properties

Triangle identity & modulus

|z₁ + z₂| < |z₁| + |z₂|,
|z| = sqrt( x² + y² ) OR |z|² = z * z̄

Polar form

z = r ( cosθ + isinθ ),

x = r cosθ and y = r sinθ

Polar form *, /, 1/z, zⁿ, z^1/n

Circles

|z - z₀| = r, where z₀ is centre, r is radius

Straight line

Im(z) = mRe(z) + c

Annulus

r₁ < |z - z₀| < r₂,

annulus is region between two cocentric circles

Sector

Complex exponential function

Exponential form

also a polar form

Complex logarithmic function

|z| is modulus of z

Logarithmic laws apply

Complex logarithmic in exponential function

Capital Ln(z) and Arg(z)

Uses only principal argument: -π < θ < π, so no “+ 2πn”