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 in the linear systems time domain.

Laplace Transform Calculator with Steps

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

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. To calculate the fourier transform function following formula is used,

$$ \int_{−∞}^∞ ∣ f(t) ∣ dt < ∞ $$

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.

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). To calculate the inverse laplace transform, following formula is used:

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

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

Amanda Jepson

Last Updated February 21, 2024