## Introduction to Laplace Transform Calculator:

Laplace transform calculator with steps is the best online source for **finding the complex periodic** function by transforming it into the frequency domain. This tool is specially designed to solve the complex differential equation in the linear systems time domain.

Laplace calculator is a beneficial tool for students, teachers, and professionals as it helps to solve various mathematical problems.

## What is Laplace Transform?

Laplace Transform is a process in which a **periodic function** is transformed into a frequency domain in complex analysis. In this way, it simplifies and analyzes the system of linear equations or differential equations.

Laplace Transform is defined as an integral of a periodic function that is multiplied by a decaying exponential function over a bounded region.

## Formula Used for Laplace Transform:

The **laplace transform formula** consists of a function f(t) in a time domain, a decay exponential function e^{-st}, 0 to ∞ lower and upper limit of a periodic function. The result is transformed from time domain function f(t) to F(s) frequency domain function. The formula used by the piecewise laplace transform calculator is given below,

$$ \mathcal{L} \{f(t) \} \;=\; \int_0^{\infty} f(t)e^{-st} dt $$

## How to Calculate Laplace Transform?

Laplace transform function is complicated to solve as it involves a **complex value function** and a real value function. If you follow our instruction then you will be able to understand how to find the Laplace transform function in steps.

**Step 1**:

Identify the given time domain function f(t) of Laplace transform.

**Step 2**:

Put the function f(t) into the laplace transform formula.

**Step 3**:

Take the integration of a given function (you can use any method of integration).

**Step 4**:

Apply limit and then simplify that expression.

**Step 5**:

After simplification, you get the solution of the given periodic function into the frequency domain.

## Example of Laplace Transform:

The laplace transform calculator with steps helps to solve laplace transform functions but its crucial to calculate such problems manually so here is an **example**,

### Example: Find the laplace to transform,

$$ f(t) \;=\; e^{-at} $$

**Solution**:

When you put the f(t) value, the laplace transform integral equation becomes,

$$ \mathcal{L} \{e^{-at} \} \;=\; \int_{0}^{\infty} e^{-st} e^{-at} dt $$

$$ \int_0^{\infty} e^{-st} e^{-at} dt \;=\; \int_0^{\infty} e^{-(s+a)t} dt $$

Take the integration with respect to t and apply limits,

$$ \int_0^{\infty} e^{-(s+a)t} dt \;=\; \biggr[ \frac{e^{-(s+a)t}}{-s(s+a)} \biggr]_{t=0}^{\infty} $$

Therefore the solution is,

$$ \left[ \frac{e^{-(s+a)t}}{-s(s+a)} \right]_{t=0}^{\infty} \;=\; \frac{1}{s+a} $$

### Example: Find the laplace transform of a given function,

$$ f(t) \;=\; t $$

**Solution**:

The laplace transform equation after adding f(t) = t value,

$$ \mathcal{L}(t) \;=\; \int_0^{\infty} e^{-t} tdt $$

Use the integration-by-parts method to get the solution of the given integral,

$$ =\; \left[ \frac{-te^{-st}}{s} \right]_{t = 0}^{\infty} + \frac{1}{s} \int_0^{\infty} e^{-st} dt $$

Apply limit and simplify it,

$$ =\; 0 + \frac{1}{s} \left[ \frac{e^{-st}}{-s} \right]_{t = 0}^{\infty} $$

$$ =\; \frac{1}{s^2} $$

## How to Use the Laplace Transform Calculator?

The laplace transformation calculator has a user-friendly design that enables you to easily calculate laplace transform questions. Before adding the input function, you just need to follow some simple steps. These steps are:

- Enter the required Laplace function in the input field.
- Add the given Laplace function variable.
- Add the variable in which you want to transform your Laplace function.
- Recheck your input function value before hitting the calculate button of laplace transform piecewise calculator.
- Click on the “
**Calculate**” button to get the desired result of your given laplace transform question. - If you are trying our laplace transform step function calculator for the first time then you should use load example option.
- Click on the “Recalculate” button to get a new page for solving laplace transform problems.

## Final Result of Laplace Calculator with Steps:

When you add an input value, the piecewise laplace transform calculator gives you the **solution** of the given problem. It provides you with the solutions which contains:

**Result Option:**

When you click on the result option then it provides you with a solution for the laplace transform question.

**Possible Step:**

When you click on the possible steps option, it provides you with the solution of laplace transform function questions in steps.

## Advantages of Laplace Transformation Calculator:

The laplace calculator with steps has many advantages that you obtain whenever you use it for calculating the Laplace transform problems. These advantages are:

- The laplace transform piecewise calculator is an
**adaptable tool**that can be opened on all electronic devices (like laptops, computers, mobile phones, tablets, etc) with the help of internet. - It is a free tool so you can find the Laplace function problems with solution without spending anything.
- Our tool saves the time and effort that you consume in doing lengthy and complex calculations of the laplace transform function.
- It is a learning tool so you can use the laplace transform step function calculator for practicing complex questions.
- Laplace transform calculator with steps is a trustworthy tool that provides you accurate solutions as per your input.
- It provides you solutions with a complete step-by-step process for a better understanding of Laplace transform problems