## Introduction to Laurent Series Calculator:

Laurent series Calculator is a best online tool that helps you to **find the complex analysis** function using laurent series expansion method. Our tool is used to simplify the process of finding complex function f(z) around a point z = z0 which is near to pole or singular point.

If you solve the laurent series of complex function by hand, you consume alot of time in getting solution or you may not get the desired result because of its complicated procedure. That’s why we introduce our laurent expansion calculator.

## What is Laurent Series?

The laurent series is a complex analysis process that use to find the **infinite sum series** of a function f(z) in z variable. It is used to describe a function f(z) that has a pole and singular point around an initial point z0.

This type of series is similar to taylor series but its function consist of a principle part and anlaytic part of a complex function. The Laurent series is implement only if the function has a convergence of radius r < ∣z−z0∣ < R, where r and R are radii of convergence around the point z0.

## Formula of Laurent Series:

The **formula of laurent series** of a complex function f(z) at point z0 used by the laurent series calculator is given as,

$$ f(z) \;=\; \sum_{n=1}^{\infty} \frac{b_n}{(z - z_0)^n} + \sum_{n=0}^{\infty} a_n (z - z_0)^n $$

Whereas,

f(z): complex variable function

bn: the analytic function

an: the principle of function f(z)

(z - z0): the difference between initial point z_0 and complex analysis function z.

## How to Find the Laurent Series?

For the **calculation** of the Laurent series of a complex function f(z) around a point z0 that involves simple steps. Here’s a method to find the Laurent series problem. These steps are

**Step 1**:

Determine where the function f(z) that is analytic and it has singularities points such as poles or essential singularities.

**Step 2**:

If f(z) has a various type of function (e.g., exponential function, trigonometric functions, rational functions),so you can get the solution of series expansions like Taylor series to express f(z) in terms of powers of z - z0.

**Step 3**:

In order to find the coefficients xc in the Laurent series then use the formula used by laurent series expansion calculator. $$ C_n \;=\; \frac{1}{2 \pi i} \int_r \frac{f(z)}{(z - z_0)^{n + 1}} dz $$

**Step 4**:

The Laurent series converges within the region r < ∣z−z0∣ < R.

## Practical Example of Laurent Series:

The laurent series calculator gives solution in steps but it’s important to understand the manual step by step calculation process. These steps are given below,

### Example: Calculate the laurent series for the following:

$$ f(z) \;=\; \frac{z + 1}{z^3(z^2 + 1)} $$

On the region A: 0 < |z| < 1 centered at z = 0.

**Solution**:

Compare the given function with the laurent series formula,

$$ f(z) \;=\; \sum_{n=1}^{\infty} \frac{b_n}{(z - z_0)^n} + \sum_{n=0}^{\infty} a_n (z - z_0)^n $$

As you can see we have only the b_n function so to find the analytic function bn use the coefficient of bn formula such as:

$$ b_n \;=\; \frac{1}{2 \pi i} \int_r f(\omega)(\omega - z_0)^{n-1} d\omega $$

First, find the singularity point to check whether function is analytic function or not and the function has isolated singularities at z = 0, ± i.

Put the value of function in the coefficient of laurent series formula,

$$ f(z) \;=\; \frac{1}{z^3}(1 + z)(1 - z^2 + z^4 - z^6 + … ) $$

Solve the above expression to get solution of given laurent series function,

$$ f(z) \;=\; \frac{1}{z^3} + \frac{1}{z^2} - \frac{1}{z} - 1 + z + z^2 - z^3 - … $$

The given laurent series is coverage as

$$ r > z_0 $$

## How to Use Laurent Series Calculator?

Laurent expansion calculator has a simple interface, so you just need to give you the laurent series problem in it to get a solution without doing anything else. Follow our guidelines before using it. These guidelines are:

**Enter the function**f(x) of the laurent series that you want to evaluate in the input box.- Add the point at which laurent series is analytic function in its respective input box.
- Check the given laurent series function before hitting the calculate button to start the evaluation process in laurent series expansion calculator.
- The “Calculate” button gives you the solution of your given Laurent series problem.
- If you are trying our laurent series online calculator for the first time then it must be suggested to try out the load example to check accuracy.
- The “Recalculate” button of our calculator that will bring you back to a new page for finding more example solutions of laurent series problems.

## Outcome from Laurent Expansion Calculator:

Laurent series Calculator gives you the **solution** to a given laurent series question when you add the input into it. It may contain as:

**Result Option**:

When you click on the result option so that it gives you a solution to the laurent series problem.

**Possible Steps**:

It provides you with a solution to the laurent series problem where calculation steps are included.

## Benefist of Laurent Series Expansion Calculator:

The laurent series online calculator gives many **benefits** that you get when you calculate the laurent series of complex function solutions. These features are:

- The laurent expansion calculator is a free tool that enables you to evaluate the laurent series problems freely.
- It is a manageable tool that can solve different types of complex variable functions to find the solution using laurent series expansion.
- Our calculator helps you to get a strong hold on the laurent series function concept when you use it for practice.
- It saves the time that you consume on the calculation of the laurent series of complex function solution in a few second.
- Our calculator provides exact solutions as per your input value when you use it to evaluate the laurent series problems without any diffculty.
- Laurent series Calculator is an educational tool that helps you to learn the complex analysis function concept by solving it with laurent series expansion method.