Complex Numbers
$$e^{i\theta}=\cos \theta+i\sin\theta$$
$$e^{a+bi}=e^a(\cos b+i\sin b)$$
$$\cos \theta = \frac{e^{i\theta}+e^{-i\theta}}{2}$$
$$\sin \theta = \frac{e^{i\theta}-e^{-i\theta}}{2}$$
2. For a complex number \(z=a+bi=r(\cos\theta+i\sin \theta)\),
The modulus \(|z|=r\)
The argument \(arg(z)=\theta\)
3. Complex conjugate. For a complex number \(z=a+bi\), its complex conjugate is \(\bar z=a-bi\)
4.
$$z_1z_2=r_1r_2(\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2))$$
$$1/z=1/r(\cos(\theta)-i\sin(\theta))$$
$$z^n=r^n(\cos(n\theta)+i\sin(n\theta))$$
6. The solution for \(x^n=z=r(\cos\theta+i\sin \theta)\) is
$$x=r^{\frac{1}{n}}(\cos(\frac{\theta+2k\pi}{n})+i\sin(\frac{\theta+2k\pi}{n})), k=0,1,2…$$
If \(n\) is a natural number, there are \(n\) different solutions for \(k=0,1…n-1\)
7. We can use Euler’s formula to derive some trig-transformations, for example:
$$\cos a\cos b=\frac{(e^{ia}+e^{-ia})(e^{ib}+e^{-ib})}{4}$$
$$=\frac{e^{i(a+b)}+e^{-i(a+b)}+e^{i(a-b)}+e^{-i(a-b)}}{4}$$
$$=\frac 1 2 (\cos(a+b)+\cos(a-b))$$