Use this calculator to find the Laplace Transform of common functions of time, f(t), resulting in a function of the complex frequency s, denoted as F(s).
Select a function and enter parameters to see its Laplace Transform.
function factorial(n) {
if (n < 0) return NaN;
if (n === 0) return 1;
var res = 1;
for (var i = 2; i <= n; i++) {
res *= i;
}
return res;
}
function updateParameterVisibility() {
var functionType = document.getElementById("functionType").value;
document.getElementById("paramCGroup").style.display = 'none';
document.getElementById("paramNGroup").style.display = 'none';
document.getElementById("paramAGroup").style.display = 'none';
document.getElementById("paramBGroup").style.display = 'none';
if (functionType === "constant") {
document.getElementById("paramCGroup").style.display = 'block';
} else if (functionType === "power_t") {
document.getElementById("paramNGroup").style.display = 'block';
} else if (functionType === "exponential") {
document.getElementById("paramAGroup").style.display = 'block';
} else if (functionType === "sin" || functionType === "cos") {
document.getElementById("paramBGroup").style.display = 'block';
}
// Dirac Delta needs no parameters
}
function calculateLaplace() {
var functionType = document.getElementById("functionType").value;
var resultDiv = document.getElementById("result");
var output = "";
try {
if (functionType === "constant") {
var C = parseFloat(document.getElementById("paramC").value);
if (isNaN(C)) throw "Please enter a valid number for Constant C.";
output = "L{C} = C/sFor C = " + C + ", F(s) = " + C + "/s";
} else if (functionType === "power_t") {
var n = parseInt(document.getElementById("paramN").value);
if (isNaN(n) || n < 0) throw "Please enter a non-negative integer for n.";
var factN = factorial(n);
output = "L{tn} = n! / sn+1For n = " + n + ", F(s) = " + factN + " / s" + (n + 1) + "";
} else if (functionType === "exponential") {
var a = parseFloat(document.getElementById("paramA").value);
if (isNaN(a)) throw "Please enter a valid number for parameter 'a'.";
output = "L{eat} = 1 / (s – a)For a = " + a + ", F(s) = 1 / (s – " + a + ")";
} else if (functionType === "sin") {
var b = parseFloat(document.getElementById("paramB").value);
if (isNaN(b) || b === 0) throw "Please enter a non-zero number for parameter 'b'.";
output = "L{sin(bt)} = b / (s2 + b2)For b = " + b + ", F(s) = " + b + " / (s2 + " + (b * b) + ")";
} else if (functionType === "cos") {
var b = parseFloat(document.getElementById("paramB").value);
if (isNaN(b) || b === 0) throw "Please enter a non-zero number for parameter 'b'.";
output = "L{cos(bt)} = s / (s2 + b2)For b = " + b + ", F(s) = s / (s2 + " + (b * b) + ")";
} else if (functionType === "dirac_delta") {
output = "L{δ(t)} = 1F(s) = 1";
} else {
output = "Please select a function type.";
}
} catch (error) {
output = "" + error + "";
}
resultDiv.innerHTML = output;
}
// Initialize visibility on page load
window.onload = updateParameterVisibility;
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Understanding the Laplace Transform
The Laplace Transform is a powerful mathematical tool used extensively in engineering and physics to solve differential equations, especially in the analysis of linear time-invariant (LTI) systems. It transforms a function of time, f(t), from the time domain into a function of a complex frequency variable, s, in the complex frequency domain (or s-domain). This transformation simplifies complex operations like differentiation and integration into algebraic manipulations, making it easier to solve problems.
What is the Laplace Transform?
Mathematically, the unilateral (one-sided) Laplace Transform of a function f(t), defined for t ≥ 0, is given by the integral:
F(s) = L{f(t)} = ∫0∞ e-st f(t) dt
Where:
f(t) is the original function in the time domain.
F(s) is its Laplace Transform in the s-domain.
s is a complex variable, s = σ + jω (where σ is the real part and jω is the imaginary part).
e-st is the kernel of the transform.
The Laplace Transform exists for functions f(t) for which the integral converges, typically requiring f(t) to be of exponential order.
Why Use the Laplace Transform?
The primary advantages of using the Laplace Transform include:
Simplification of Differential Equations: It converts linear differential equations with constant coefficients into algebraic equations, which are much simpler to solve.
Initial Conditions: Initial conditions are naturally incorporated into the transformed equation, simplifying the solution process.
System Analysis: It provides insights into the stability, frequency response, and transient behavior of dynamic systems (e.g., electrical circuits, mechanical systems).
Convolution: Time-domain convolution (which can be complex) becomes simple multiplication in the s-domain.
Common Laplace Transform Pairs
While the integral definition is fundamental, in practice, engineers and mathematicians often use tables of common Laplace Transform pairs. This calculator provides a quick way to find these common transforms:
Constant Function:L{C} = C/s
Power Function:L{tn} = n! / sn+1 (for non-negative integer n)
Exponential Function:L{eat} = 1 / (s - a)
Sine Function:L{sin(bt)} = b / (s2 + b2)
Cosine Function:L{cos(bt)} = s / (s2 + b2)
Dirac Delta Function:L{δ(t)} = 1
How to Use This Laplace Transform Calculator
Select Function Type: Choose the type of function f(t) you want to transform from the dropdown menu (e.g., Constant, tn, eat, sin(bt), cos(bt), Dirac Delta).
Enter Parameters: Depending on your selected function, relevant input fields for parameters like C, n, a, or b will appear. Enter the specific numerical values for these parameters.
Calculate: Click the "Calculate Laplace Transform" button.
View Result: The calculator will display the Laplace Transform F(s) for your chosen function and parameters in the result area.
Examples of Laplace Transforms
Let's look at a few examples using realistic numbers:
Example 1: Constant Function
If f(t) = 5 (select 'Constant', enter C=5), the calculator will show:
L{5} = 5/s
Example 2: Power Function
If f(t) = t3 (select 'tn', enter n=3), the calculator will show:
L{t3} = 3! / s3+1 = 6 / s4
Example 3: Exponential Function
If f(t) = e-2t (select 'eat', enter a=-2), the calculator will show:
L{e-2t} = 1 / (s - (-2)) = 1 / (s + 2)
Example 4: Sine Function
If f(t) = sin(4t) (select 'sin(bt)', enter b=4), the calculator will show:
L{sin(4t)} = 4 / (s2 + 42) = 4 / (s2 + 16)
Example 5: Cosine Function
If f(t) = cos(3t) (select 'cos(bt)', enter b=3), the calculator will show:
L{cos(3t)} = s / (s2 + 32) = s / (s2 + 9)
This calculator is a helpful tool for quickly verifying or finding the Laplace Transforms of these fundamental functions, aiding in your study and application of control systems, circuit analysis, and other engineering disciplines.