Complex Numbers

Conjugate, polar form, modulo.

10 flashcards · Mathematics · by Athena Papkouf

1. Let a complex number be z = a + ib. conjugate zˉ\bar{z}zˉ

   zˉ=a−ib\bar{z}=a-i bzˉ=a−ib

2. Re(z) in terms of zˉ \bar{z} zˉ

   z+zˉ2\frac{z+\bar{z}}{2}2z+zˉ​

3. Im(z) in terms of zˉ \bar{z} zˉ

   z−zˉ2i\frac{z-\bar{z}}{2 i}2iz−zˉ​

4. z+z′‾\overline{z+z^{\prime}}z+z′​

   zˉ+z′‾\bar{z}+\overline{z^{\prime}}zˉ+z′

5. zz′‾\overline{z z^{\prime}}zz′

   zˉz′‾\bar{z} \overline{z^{\prime}}zˉz′

6. modulo |z|

   ∣z∣=a2+b2|z|=\sqrt{a^{2}+b^{2}}∣z∣=a2+b2 ​

7. polar form

   z=r(cos(φ)+isin(φ))z=r(cos(φ)+isin(φ))z=r(cos(φ)+isin(φ)) === reiϕre^{i\phi}reiϕ r=∣z∣r=|z|r=∣z∣ φ=arg⁡(z)∈]−π,π]\varphi=\arg (z) \in]-\pi, \pi]φ=arg(z)∈]−π,π] cos⁡(φ)=aa2+b2,sin⁡(φ)=ba2+b2\cos (\varphi)=\frac{a}{\sqrt{a^{2}+b^{2}}}, \quad \sin (\varphi)=\frac{b}{\sqrt{a^{2}+b^{2}}}cos(φ)=a2+b2 ​a​,sin(φ)=a2+b2 ​b​

8. Formula de Moivre for powers of z

   zn=(rn,nφ)=rn(cos⁡(nφ)+isin⁡(nφ))z^{n}=\left(r^{n}, n \varphi\right)=r^{n}(\cos (n \varphi)+i \sin (n \varphi))zn=(rn,nφ)=rn(cos(nφ)+isin(nφ))

9. i2

   -1

10. ∣zˉ∣|\bar{z}|∣zˉ∣

    ∣z∣|z|∣z∣