Complex number for biologist

A biologist like me might have never had a numerical computing training. I don’t even known what a complex number really means. Here are some useful basics to keep in mind.

Complex number

complex number $a+bi$ lives in a 2d complex plane, including

• real axis: $a$
• imagnary axis: $i$
  • orthognal to real axis
  • $i \rightarrow 90 \degree \text{rotation}$

2 ways of representation

• $z = a + bi$
• $z = r \cos(\phi) + r \sin(\phi) i = r e^{i \phi}$

3 Facts about Multiplication

• $z \cdot 1 = z$
• $z \cdot i = \operatorname{Rot90}(z)$
  • e.g. $i \cdot i = -1$
• $z \cdot ( c + di) = c \cdot z + d \cdot (zi)$
  • e.g. $(2+i)(2-i) = 2 \cdot 2 + 2i -2i - i^2 = 5 + 0i$

expotential

form: $$ \exp (i \theta)=1+i \theta+\frac{(i \theta)^{2}}{2}+\frac{(i \theta)^{3}}{6}+\frac{(i \theta)^{4}}{24}+\cdots $$

derivative:

$$ \frac{d}{d t} e^{i t}=i \cdot e^{ i \cdot t} $$

$$ i^n \cdot i^k = i^{n+k} $$

expotential form to find complex roots

Example:

Reference

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