Complex Number & Complex Conjugate 複數、共軛複數

complex number 複數
complex plane 複數平面
real part 實部
imaginary part 虛部

z = a + bi
Re(z) = a
Im(z) = b
i² = -1
-i = 1/i proof: -i = -i × i/i = -i²/i =-(-1)/i = 1/i
∠z = tan^-1(Im(z)/Re(z)) = tan^-1(Ib/a)

z = r∠θ
r is modulus 模數 / magnitude 量值/大小
θ is argument 幅角/引數 / phase 相位/相角
r = |z|

argument 幅角
phase 相位/相角
in-phase 同相位
out-of-phase 不同相位
For Acos(ax + b), phase is b. and amplitude is A.

radian 徑度/弧度

phasor 相量

e^jx = cosx + isinx = cisx     (Euler's formula)

complex conjugate 共軛複數
if z = a + bi,
the complex conjugate of z is
z_bar or z* = a - bi
Re(z*) = Re(z) = a
Im(z*) = -Im(z) = -b

To get z*, simply flip z around x-axis.

|z| = | a + bi | = √(a² + b²)
|z|² = |a + bi|² = a² + b² = (a + bi )(a - bi ) = (z)(z*)

Matlab: conj()

>> z = 1 + 2i

z =
   1.0000 + 2.0000i

>> z_conj = conj(z)

z_conj =

   1.0000 - 2.0000i

conjugate transpose 共軛轉置

相關資料

Euler's formula