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🎓 Xin Yu's NoteZzz

W4 Complex Numbers

Cartesian Form

  • z=x+iy\boxed{z=x+iy}, where x,y∈Rx,y\in\mathbb{R}
    • x=Re(z)x=Re(z): reeal part of zz
    • y=Im(z)y=Im(z): imaginary part of zz
    • i2=−1i^2=-1
  • Cartesian representation:

Polar Form

  • r(cos⁡θ+isin⁡θ)\boxed{r(\cos\theta+i\sin\theta)}
    • r=∣z∣=x2+y2r=|z|=\sqrt{x^2+y^2}
    • tan⁡θ=yx\tan\theta=\frac{y}{x}
    • cos⁡θ=xx2+y2\cos\theta=\frac{x}{\sqrt{x^2+y^2}}
    • sin⁡θ=yx2+y2\sin\theta=\frac{y}{\sqrt{x^2+y^2}}

Principal Argument, P.V.arg(z)P.V. arg(z)

  • −π<P.V.arg(z)≤π-\pi<P.V.arg(z)\le\pi

Complex Exponential

  • eiθ=cos⁡θ+isin⁡θ\boxed{e^{i\theta}=\cos\theta+i\sin\theta}
  • Polar form: z=reiθ\boxed{z=re^{i\theta}}

Properties of Complex Exponential

  1. ei0=1e^{i0}=1
  2. eiθeiΦ=ei(θ+Φ)e^{i\theta}e^{i\Phi}=e^{i(\theta+\Phi)}

Products and Division in Polar Form

  • If z=r1eiθz=r_1e^{i\theta} and w=r2eiΦw=r_2e^{i\Phi} then:
  1. zw=r1r2ei(θ+Φ)zw=r_1r_2e^{i(\theta+\Phi)}
  2. zw=r1r2ei(θ−Φ)\frac{z}{w}=\frac{r_1}{r_2}e^{i(\theta-\Phi)}

De Moivre’s Theorem

  • zn=(reiθ)n=rneinθz^n=(re^{i\theta})^n=r^ne^{in\theta}
    • z=reiθz=re^{i\theta}
    • nn is a positive integer

Exponential Form of sin⁡θ\sin\theta and cos⁡θ\cos\theta

  • cos⁡θ=12(eiθ+e−iθ)\cos\theta=\frac{1}{2}(e^{i\theta}+e^{-i\theta})
  • sin⁡θ=12i(eiθ−e−iθ)\sin\theta=\frac{1}{2i}(e^{i\theta}-e^{-i\theta})
  • connection with hyperbolic trigonometric functions:
    • cosh⁡(iθ)=12(eiθ+e−iθ=cos⁡θ\cosh(i\theta)=\frac{1}{2}(e^{i\theta}+e^{-i\theta}=\cos\theta
    • sinh⁡(iθ)=12i(eiθ−e−iθ=sin⁡θ\sinh(i\theta)=\frac{1}{2i}(e^{i\theta}-e^{-i\theta}=\sin\theta

Differentiation via Complex Exponential

Integration via Complex Exponential