Calculator Inputs
This page numerically evaluates selected integral transforms over user-defined bounds. Use explicit multiplication such as 3*x, x*sin(x), or exp(-2*x).
Example Data Table
These examples illustrate common numerical settings and expected approximate outputs.
| Transform | Function | Parameter | Bounds | Amplitude | Scale | Approximate Output |
|---|---|---|---|---|---|---|
| Laplace | 1 | s = 2 | 0 to 10 | 1 | 1 | 0.4999999990 |
| Complex Fourier | exp(-x) | ω = 1 | 0 to 12 | 1 | 1 | 0.5 - 0.5i |
| Fourier Sine | exp(-x) | ω = 1 | 0 to 12 | 1 | 1 | ≈ 0.5 |
| Fourier Cosine | exp(-x) | ω = 1 | 0 to 12 | 1 | 1 | ≈ 0.5 |
| Mellin | exp(-x) | s = 2 | 0.001 to 12 | 1 | 1 | ≈ 1 |
Formula Used
General Scaled Integrand
g(x) = A · f(kx), where A is the amplitude factor and k is the variable scaling factor.
Laplace Transform
L(s) = ∫ab A·f(kx)e-sx dx
Complex Fourier Transform
F(ω) = ∫ab A·f(kx)e-iωx dx, with real part from cosine and imaginary part from negative sine.
Fourier Sine and Cosine Transforms
Fs(ω) = ∫ab A·f(kx)sin(ωx) dx
Fc(ω) = ∫ab A·f(kx)cos(ωx) dx
Mellin Transform
M(s) = ∫ab xs-1A·f(kx) dx. The selected interval must stay positive.
Numerical Method
The calculator uses composite Simpson’s rule. Accuracy generally improves with smoother functions, better bounds, and higher even interval counts.
How to Use This Calculator
1. Select a transform
Choose Laplace, complex Fourier, Fourier sine, Fourier cosine, or Mellin from the transform menu.
2. Enter the function
Type the input as f(x). Use explicit multiplication, for example 2*x, x*sin(x), or exp(-x).
3. Set parameter and bounds
Enter the transform parameter and define the lower and upper bounds for numerical integration.
4. Adjust amplitude and scaling
Use amplitude A to weight the function and scale k to evaluate the transformed version A·f(kx).
5. Choose interval count
Higher interval counts often improve accuracy for oscillatory or rapidly changing functions.
6. Submit and export
Press the calculate button to show results above the form, then download the output as CSV or PDF.
FAQs
1. What is an integral transformation?
An integral transformation rewrites a function by integrating it against a kernel. It helps convert problems into forms that are easier to study, simplify, or solve numerically.
2. Which transforms are available here?
This calculator supports Laplace, complex Fourier, Fourier sine, Fourier cosine, and Mellin transforms over user-selected numerical bounds.
3. Are the results exact or approximate?
The outputs are numerical approximations computed with composite Simpson’s rule. Better bounds and higher interval counts usually improve the estimate.
4. What function syntax should I use?
Use x as the variable and write explicit multiplication, such as 3*x or x*sin(x). Supported functions include sin, cos, exp, log, ln, sqrt, abs, and more.
5. Why does interval count matter?
More intervals usually capture oscillation and curvature better. That is especially helpful for Fourier kernels or sharply changing functions.
6. Why must Mellin bounds stay positive?
The Mellin transform uses x^(s−1). Intervals touching or crossing zero can make the kernel undefined or singular, so only positive bounds are allowed.
7. What do amplitude and scale change?
Amplitude multiplies the full function, while scale changes the input variable through f(kx). This helps model weighted, stretched, or compressed inputs.
8. What do the export buttons save?
CSV export saves the summary values in tabular form. PDF export captures the displayed result card for sharing, archiving, or printing.