## Introduction to Puiseux Series Calculator:

Puiseux series calculator is an online source that helps you to **find the puiseux series** of a given function at a specific point where it may coverage or diverge. It is used to evaluate the singularity point from the series of a function that has fractional power of x variable.

The Puiseux Series Finder is a valuable tool as it gives you solution of complex function variable into puiseux series of expansion in a few second without doing manual calculation easily.

## What is Puiseux Series?

A Puiseux series is an advance complex analysis method which is used to find the attribute of a fractional **exponential power function** around a specific point. It is a generalization of a power series but in the fractional powers of the variable x.

This type of series expansion is particularly useful in complex variable function and algebraic geometry, where it is used to evaluate the convergence of a functions that have singularities points.

## Puiseux Series Expansion:

The **puiseux series expansion** has a function f(x) around a point x = x0, in which an is the coefficient value and pn /qn is the fractional power over the series and n is the positive integer that value is varies. The puiseux series expansion by the Puiseux Series Calculator is given below,

$$ f(x) \;=\; \sum_{n=0}^{\infty} a_n (x - x_0)^{\frac{pn}{qn}} $$

## How to Find Puiseux Series?

To **calculate a Puiseux series** you need to expand the series to find the coefficients and exponential power from the given function around a given point x = x_0. Here's a step-by-step guide to calculate a Puiseux series:

**Step 1**:

Determine a function f(x) that you want to expand into a Puiseux series at a point x = x0.

**Step 2**:

Put the point value in the formula of Puiseux series expansion that is given as,

$$ f(x) \;=\; \sum_{n=0}^{\infty} a_n (x - x_0)^{\frac{pn}{qn}} $$

**Step 3**:

Find the value of the function f(x) near x = x0. This term will give you the initial coefficient a0and exponent power p0/q0.

**Step 4**:

To find the subsequent coefficients an and exponents pn/qn as n = 0,1,2,3…, iteratively the Puiseux series form into the function f(x) and equating coefficients of like powers of (x - x0).

**Step 5**:

After solving the above equations to get the coefficients an and pn/qn for each n. Put these value in the puisex series formula to make it a series.

**Step 6**:

Lastly, check the convergence of the series within the radius point x0.

## Practical Example of Puiseux Series:

The practical **example** of a Puiseux series with solution is given below to let you know about the working process of Puiseux Series Calculator.

### Example: Find the Puiseux series expansion of the following:

$$ f(x) \;=\; \sqrt{1-x}\; around\; x \;=\; 0 $$

**Solution**:

Identify the given function and the particular point value,

$$ f(x) \;=\; \sqrt{1-x}\; x \;=\; 0 $$

The puisex series expansion at a point x = 0 is,

$$ f(x) \;=\; \sum_{n=0}^{\infty} a_n (x - 0)^{\frac{pn}{qn}} $$

As we have x = 0 so find the value of f(x0) by putting it into f(x),

$$ a_0 \;=\; f(0) \;=\; \sqrt{1-0} \;=\; 1 $$

And put x=0 in the fractional power of series,

$$ \frac{p_0}{q_0} \;=\; \frac{1}{2} $$

Expand the puiseux series on the given function f(x),

$$ \sqrt{1-x} \;=\; (1 - x)^{\frac{1}{2}} $$

To find the value of a1 and p1/q1 you need to expand the above function into series,

$$ (1 - x)^{\frac{1}{2}} \;=\; 1 - \frac{1}{2}x - \frac{1}{8}x^2 - \frac{1}{16}x^3 - … $$

$$ a_1 \;=\; −\frac{1}{2}\; and\; \frac{p_1}{q_1} \;=\; \frac{1}{2} $$

To find the a2 and p2/q2 value, again expand the series

$$ (1 - x)^{\frac{1}{2}} \;=\; 1 - \frac{1}{2}x - \frac{1}{8}x^2 - \frac{1}{16}x^3 $$

Where,

$$ a_2 \;=\; -\frac{1}{8}\; and\; \frac{p_2}{q_2} \;=\; \frac{1}{2} $$

So on you need to find the nth value for pisuex series expansion.The result after adding the value of an and fractional power is given as,

$$ (1 - x)^{\frac{1}{2}} \approx 1 - \frac{1}{2}x - \frac{1}{8}x^2 $$

## How to Use Puiseux Series Calculator?

The Puiseux Series Finder has a user-friendly layout that allows you to solve various types of complex function around a singular point problems. You just need to put your problem in this calculator and follow some guidelines which are given as:

- Add the singular point of puiseux series function in the input field.
- Enter the given function of puiseux series in the input field.
- Recheck your given input puiseux series function before clicking on the calculate button to get the solution in the form of puiseux series expansion.
- Click on the “
**Calculate**” button for the solution puiseux series problems. - If you want to check the working process of the Calculator Puiseux Series then use the load example for the calculation of series
- The “Recalculate” button allows you to evaluate more examples of puiseux series expansion problem with solution.

## Final Result of Puiseux Series Finder:

Puiseux series Calculator provides you with a **solution** of puiseux series of complex function as per your input when you click on the calculate button. It may include as:

**In the Result Box**:

When you click on the result button you get the solution of puiseux series problem

**Steps Box**:

Click on the steps option so that you get the solution of puiseux seris questions in a step-by-step method.

## Advantages of Using Calculator Puiseux Series:

The Puiseux Series Finder has several advantages that you obtain whenever you use it to solve puiseux series problems to get the solution. Our tool only gets the input value and gives solution of puiseux series expansion without any restriction. These advantages are:

- Our calculator is a
**trustworthy tool**as it always provides you with accurate solutions of puiseux series problem. - It is a swift tool that evaluates puiseux series problems with solutions in a couple of seconds.
- It is a learning tool that helps children about the concept of puisex series of a given function very easily on an online platforms.
- It is a handy tool that solves puieseux series problems quickly because you do not give any type of asistance for calculation.
- Puiseux series Calculator is a free tool that allows you to use it for the calculation of puiseux series without getting any fee.
- It is an easy-to-use tool, anyone or even a beginner can easily use it for the solution of puiseux series problems.
- It can operate on a desktop, mobile, or laptop through the internet to solve puiseux series problems.