Laplace Transform Calculator

The laplace transform calculator is a helpful tool for finding the complex periodic function by transforming it into the frequency domain for free.

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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 into linear systems in the time domain.

Laplace Transform Calculator with Steps

Laplace calculator is a beneficial tool for students, teachers, and professionals as it is used in various fields of mathematics such as physics, engineering, mathematics, etc.

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 ∞ is the 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 has a complicated calculation because it involves a complex value function and a real value function. If you follow our instruction that tells you how to find the Laplace transform function in steps then you get a clear idea about its mathematical procedure.

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 such as

Step 3:

Take the integration of a given function (you can use any method of integration to get a solution.

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.

Solved Example of Laplace Transform:

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

Example: Find the laplace to transform,

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

Solution:

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

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

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

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 use it to easily calculate laplace transform questions. Before adding the input function to get solutions, you just need to follow some simple steps. These steps are:

Enter the required Laplace function that you want to evaluate 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 to start the calculation process in the laplace transform piecewise calculator.
Click on the “Calculate” button to get the desired result of your given laplace transform questions with a solution
If you want to try out ourlaplace transform step function calculator for the first time then you can give different examples to see the working method and its accuracy.
Click on the “Recalculate” button to get a new page for solving Laplace Transform problems to get solutions

Final Result of Laplace Calculator with Steps:

The piecewise laplace transform calculator gives you the solution to a given problem when you add the input value in it. It provides you with the solutions which may contain as:

Result Option:

You can click on the result option and 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 to laplace transform function problems in steps.

Advantages of Laplace Transformation Calculator:

The laplace calculator with steps has many advantages that you obtain whenever you use it for the calculation of Laplace transform problems and get solutions without giving any manual instruction. These advantages are:

The laplace transform piecewise calculator is an adaptable tool that can open on all electronic devices like laptops, computers, mobile, tablets, etc with the help of the internet.
It is a free tool so you can use it to find the Laplace function problems with a solution freely without spending anything
Our tool saves the time and effort that you consume in doing lengthy and complex calculations of the laplace transform function in a few seconds.
It is a learning tool so you can use the laplace transform step function calculator for practice so that you get in-depth knowledge.
It is a trustworthy tool that provides you with accurate solutions according to your input whenever you use the laplace transform calculator with steps to get a solution.
It provides you solutions with a complete process in a step-by-step method for a better understanding of Laplace Transform problems

Related References

Frequently Ask Questions

What is the difference between Laplace and Fourier transform

Laplace transform and Fourier transform are both complex analysis methods which is used to analyze functions and differential equations. The difference between the laplace and Fourier transform functions:

Fourier Transform functions are defined as functions that transform a function into a frequency domain (denoted as ω or k) from the periodic domain. It is commonly used in areas such as signal processing, optics, and quantum mechanics to analyze the frequency of a signal or wave.

It converges if the function is integrable over its domain

$$ \int_{−∞}^∞ ∣ f(t) ∣ dt < ∞ $$

Laplace Transform functions are defined as functions that change the complex variable function into the time domain from the frequency domain in the half-open interval [0,∞). It analyzes signals and systems in the context of engineering and physics, where initial conditions (at t=0) are specified. It requires exponential decay of the function as t goes to infinity. Laplace transform exist if

$$ \int_0^∞ e^{−σt} ∣ f(t) ∣ dt < ∞ \;for\; some\; σ > 0 $$

In short, both transforms involve converting a function from one domain to another to simplify the analysis of different types of functions that involved over a specific condition.

What is Laplace transform used for

Laplace transform is a method that is used in engineering, physics, and applied mathematics for solving differential equations, analyzing linear time-invariant systems, and understanding complex behaviors of function over time. Here are some specific applications of the Laplace transform:

Laplace transform allows differential equations (both ordinary and partial) to be transformed into algebraic equations, which are easier to solve it
In engineering disciplines such as electrical, mechanical, and control systems, It is used to analyze the behavior of linear time-invariant systems in studying the response of various inputs, stability analysis, frequency, and control systems.
In communication systems it is used in signal processing, such as filtering, modulation, and effects of noise and interference on signals.
In probability theory and statistics, the Laplace transform is used to find moment generating functions of random variables, which are useful in probability distributions.

Can you multiply laplace transforms

Yes, you can multiply the Laplace transforms over certain conditions which are given as:

Linearity of the Laplace Transform:

Laplace transform is a linear operation that allows you to multiply a function of a constant before taking the Laplace transform.

$$ L [f(t)] \;=\; F(s) \;and\; L [g(t)] \;=\; G(s) $$

Then for constants a and b:

$$ L [af(t) + bg(t)] \;=\; aF(s) + bG(s) $$

Convulation Theorem:

It is a specific theorem for multiplying Laplace transforms of functions which is convolution integral.

$$ L [f(t)] \;=\; F(s)\; and\; L [g(t)] \;=\; G(s) $$

Then:

$$ L [f(t) . g(t)] \;=\; \int_0^t f (τ) g(t - τ) d τ \;=\; F(s) * G(s) $$

Where “∗” denotes the convolution operation in the Laplace transform domain.

Multiplication of Transformed Functions:

If you have two functions f(t) and g(t), and their Laplace transforms F(s) and G(s), respectively, you can multiply their Laplace transforms F(s)⋅G(s) to obtain the Laplace transform of their product f(t)⋅g(t).

In short, while you can multiply Laplace transforms under appropriate conditions, the result in the Laplace domain remain in the time domain. This property makes Laplace transforms particularly useful for solving differential equations and analyzing linear systems.

What is the laplace transform of sint

To find the Laplace transform of sin⁡(t), denoted as L{sin⁡(t)}, we use the definition of the Laplace transform:

$$ L [sin (t)] \;=\; \int_0^∞ e^{-st} sin(t)\; dt $$

Where s is a complex variable parameter in the Laplace transform.

Take the integration with respect to t,

$$ L [sin (t)] =\;=\; \int_0^∞ e^{-st} sin(t) dt $$

We can use the identity for the Laplace transform of sin⁡(t),

$$ L [sin (t)] \;=\; \frac{1}{s^2} + 1 $$

This result is valid for Re(s)> 0, ensuring the convergence of the integral.

Therefore, the Laplace transform of sin⁡(t)is 1/s^2+1.

What is inverse laplace transform

The inverse transform refers to the process of reversing a mathematical transformation where a function converts from the transformed domain (such as the complex s-domain for Laplace transforms or the frequency domain for Fourier transforms) into its original domain.

Inverse Laplace Transform:

The Laplace transform is defined as:

$$ F(s) \;=\; L [f(t)] \;=\; \int_0^∞ e^{-st} f(t) dt $$

The inverse Laplace transform is denoted as L^−1{F(s)} and it retrieves f(t) from F(s). It is given by:

$$ f(t) \;=\; L^{-1} [F(s)] \;=\; \frac{1}{2πi} \int_{⋎ + i∞}^{⋎-i∞} e^{st} F(s)\; ds $$

Where γ is chosen such that F(s) converges for Re(s) > γ.

Amanda Jepson

Last Updated February 21, 2024