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# Residual Calculator + Online Solver With Free Steps

The **Residual Calculator **is an advanced online tool that helps to find the residue of any mathematical function. The **residue** is the coefficient of the first term with negative power in **Laurentâ€™s** series of functions about a point.

The **calculator** takes the information regarding the target function and point as the input and returns the residue of the provided function.

## What Is the Residual Calculator?

**The Residual Calculator is an online calculator that can calculate the residue of any complex function with the speed of knots.**

It becomes challenging to deal with these calculations if solved manually. Therefore, we offer you the **Residual Calculator** that solves your problem by quickly determining the residue.

This tool’s high accuracy and precision make it reliable and efficient. It is a powerful tool for helping mathematicians and students find solutions easily and quickly.

To find the **residue**, you first need to find Laurent’s series and then extract the coefficient from the resultant expression at the given point. This method involves numerous calculations and approximations.

## How To Use the Residual Calculator?

You can use the **Residual Calculator** by entering your function’s expression and a point in the provided space.Â You can follow the steps given below to use the calculator correctly.

### Step 1

Put the **expression** of the function for which you want to solve the problem in the â€˜**Residue of**â€™ box.

### Step 2

Insert the target **point** where you want to calculate the residue in the same field and separate it with a comma. It is actually a pole of the complex function.Â

### Step 3

Press the â€˜**Submit**â€™ button to get the solution.

### Result

The result of the calculator varies on the expression provided at the input. If the pole is defined explicitly in the input, then it gives the resultant **residue** at the pole.

If the pole is not provided, then the calculator finds the **pole** and displays the value of the pole and the residue at the pole. Also, it gives the representation of the zeros and poles of the function in a complex plane.

## How Does the Residual Calculator Work?

The residual calculator works by finding the** residue **of a given function. It calculates the residue value at a **simple** pole and a **higher order** pole of a process. It also provides the pole-zero plot.

The usage of this calculator will be cleared by knowing the concept behind the **residue **value and the poles of a function.

### What Is Residue?

The** residue **of a function â€˜**f**â€™ at a value â€˜**c**â€™ is the **coefficient** $b_{-1}$ of the first **negative **power term in the Laurent series expansion of function â€˜**f**â€™ at constant â€˜**c**.â€™

It can be defined mathematically by considering the **isolated singular** function â€˜**f(z)**â€™ at $z_{0}$ with the **Laurent series **expansion of â€˜**f(z)**â€™ given by:

\[f(z) = \displaystyle\sum_{n=0} ^{\infty} a_n{(z-z_o)^n} + \displaystyle\sum_{n=1} ^{\infty} \frac{b_n}{(z-z_o)^n}\]

In the above expansion, the** residue **of â€˜**f(z)**â€™ at $z_{0}$ is the coefficient $b_{1}$ which is represented by $Res_{(z=z_{0})} f$ = $b_{1}$.Â

The above residue is defined for the function that has isolated singularity at $z_{0}$.

The **isolated singular **point $z_{0}$ of a function is that point in the complex plane for the function is **defined** and** differentiable **at every point except the point $z_{0}$ itself.

### How To Find the Residue Value?

The residue value can be found through the** Laurent series** since the coefficient of the first **negative** term in the expansion represents its value. However, it is a tedious task.

The simple and standard method to determine the residue is through the** formulae** that find the residue at the** poles **of the given function. The calculation of the residues at the poles made the task very easy.

#### Residue at Simple Pole

The pole of the function with power equal to **one** is known as a** simple pole**. The limit formula gives the residue at a simple pole, and it is defined below.

If a function â€˜**f(z)**â€™ has a simple pole at $z_{0}$ and it has an isolated singularity at $z_{0}$, then the** residue** at that simple pole is given by:

\[Res_{(z=z_{0})} = b_{1} =Â \lim_{z\to z_{0}} (z-z_{0}) f(z) \]

In the above formula, if the limit does not exist, thereÂ is no isolated singularity at $z_{0}$. If the limit is equal to** zero, there is a removable singularity, and if the pole is of a higher order,**Â then the limit results in** infinity**.

#### Residue at Higher Order Pole

The formula also gives the residue at higher order poles as at the simple poles.

If a function â€˜**f(z)**â€™ has a pole of order â€˜**m**â€™ at $z_{0}$ and the isolated singularity for the function exists at $z_{0}$, then the **residue** is calculated using the following formula:

\[Res_{(z=z_{0})} = b_{1} = \frac{1}{(m-1)!} \, \lim_{z\to z_{0}} \frac{d^{(m-1)}}{d\,z^{(m-1)}} ((z-z_{0})^m f(z)) \]

This formula is very handy in finding the residue for the poles that have low power; however, for the higher power poles, it becomes very complex. Therefore, in that case, finding the residue through Laurent series expansion becomes easier.

## Solved Examples

Letâ€™s solve problems using the **Residual Calculator** to enhance the concepts and working of the calculator.

### Example 1

A feedback-based control system has the following transfer function.

\[ f(z) = \dfrac{z^2 + 2}{z + 3} \]

Find the poles for this function and residue at the resultant bars.

### Solution

The calculator gives the following results for the above problem.

#### Residue

The residue for the transfer function is given below.

\[\underset{z=-3}{\text{Res}} \left(\frac{z^2 + 2}{z + 3} \right) = 11\]

#### Poles

The given function has only one pole, which is given below.

**z = -3** (simple pole)

#### Pole-Zero Plot

Figure 1 shows the pole and zeros for the given function plotted in a complex plane.

### Example 2

Consider the function given function:

\[ f(z) = \frac{\cos{(z)}}{z^4\, – \,1} \]

Find the residue of the function at point **z=i**.

### Solution

#### Residue

Now the point is defined; therefore, the calculator only returns the value of residue at this point, which is given as follows:

\[ \underset{z=i}{\text{Res}} \left(\frac{\cos{z}}{z^4\, -\, 1} \right) = \frac{1}{4}i\cosh{(1)} \]