#algebra 

The Laurent series of a [[Комплексные числа|complex]] function $\LARGE f(z)$ is a power series which includes terms of negative degree

Often used to express complex functions in cases where a [[Ряд Тейлора|Taylor series]] expansion cannot be applied

## Why it's useful
![[Pasted image 20251028221133.png]]
Say you have a disk centered at $\LARGE z_1$, with the closest singularity point being $\LARGE z_2$, so $\LARGE R=|z_2-z_1|$ 
However, if we wanted to build a series that would converge at point x, Taylor series expansion wouldn't be appropriate here.
To do so, one must incorporate negative powers


## Formal definition
![[Pasted image 20251028221356.png]]
Gray area - $\LARGE f(z)$ is [[Holomorphic function|analytic]]

By [[Cauchy's integral formula]]:
$$\LARGE f(a+h)=\frac{1}{2\pi i}\oint_C\frac{f(z)}{z-a-h}dz-\frac{1}{2\pi i}\oint_{C'}\frac{f(z)}{z-a-h}dz$$
We want to represent each integral as a power series:
First one:
![[Pasted image 20251028222129.png]]
(DO NOT WRITE THOSE AS DERIVATIVES OF OUR FUNCTION, THEY DON'T EXIST THERE AND THE VALUE OF THE INTEGRAL IS DIFFERENT)

For the second part:
![[Pasted image 20251028222159.png]]
![[Pasted image 20251028222207.png]]

Therefore,
$$\LARGE f(a+h)=\sum_{n=0}^\infty a_nh^n +\sum_{n=1}^\infty \frac{b_n}{h^n}$$
where
$$\LARGE b_n=\frac{1}{2\pi i}\oint_{C'}f(z)(z-a)^{n-1}dz$$

The first part of the Laurent expansion is called its **regular** (правильная) part
The second - **principal** (главная) part

### Example
![[Pasted image 20251028222601.png]]
(in $\LARGE a_n$ the imaginary part is 0 cuz odd)
![[Pasted image 20251028223053.png]]

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Laurent series of a fraction with a polynomial in its denominator:
![[Pasted image 20251128141239.png]]
![[Pasted image 20251128141307.png]]

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![[Pasted image 20251128141642.png]]