Calculator Inputs
Choose a mode, enter the required values, and submit to display the result above this form.
Example Data Table
| Case | Inputs | Key Identity | Sample Output |
|---|---|---|---|
| Point approximation | x = 1, a = 1, σ = 0.2, W = 1 | δσ(x-a) = exp(-(x-a)²/σ²)/(σ√π) | Approximate value ≈ 2.820948 |
| Scaled impulse | x = 2, k = 2, a = 4, σ = 0.25, W = 3 | δ(kx-a) = (1/|k|)δ(x-a/k) | Support = 2, area = 1.5 |
| Sifting polynomial | L = -2, U = 2, a = 1, W = 2, f(x)=1+x² | ∫f(x)Wδ(x-a)dx = Wf(a) | Integral = 4 |
| Sifting outside interval | L = 0, U = 1, a = 3, W = 5 | Outside support gives zero | Integral = 0 |
Formula Used
δσ(x − a) = exp(−(x − a)² / σ²) / (σ√π)
W·δσ(x − a) gives the same total area W.
δ(kx − a) = (1 / |k|)·δ(x − a/k), for k ≠ 0
∫LU f(x)·W·δ(x − a) dx = W·f(a), when a is inside [L, U]
The Dirac delta is not an ordinary function. This calculator uses a narrow Gaussian to provide practical numeric estimates while preserving the core impulse identities used in mathematics, physics, and engineering.
How to Use This Calculator
- Select the mode that matches your problem.
- Enter the support point, weight, and any evaluation values.
- Use a smaller σ for a sharper impulse approximation.
- For scaled impulses, supply k and the calculator reduces the support automatically.
- For integrals, choose a function type and fill its coefficients.
- Press the calculate button to show the result above the form.
- Download the result as CSV or PDF when needed.
Frequently Asked Questions
1) What does this calculator actually compute?
It computes practical numeric approximations and identities involving the Dirac delta, including point values, scaled forms, and sifting integrals over selected intervals.
2) Why is a Gaussian width σ required?
The true Dirac delta is a distribution, not a regular function. The Gaussian width creates a narrow approximation that can be evaluated numerically.
3) What happens when σ becomes very small?
The approximation becomes sharper and taller near the support point. Numerical values increase near the center while the total area stays controlled.
4) How does the scaled identity help?
It rewrites δ(kx − a) into a standard shifted delta with a weight adjustment of 1/|k|. This makes support points and total area easier to interpret.
5) Why can an integral become zero?
If the support point a lies outside the integration interval, the impulse does not contribute inside that range, so the sifting integral returns zero.
6) Which function types can I use for f(x)?
This version supports quadratic polynomials, sine functions, cosine functions, and exponential expressions with adjustable coefficients for quick analysis.
7) Is this tool useful for physics problems?
Yes. Dirac delta models impulses, idealized point sources, sudden forcing, Green’s functions, and sampling behavior in many physics and engineering contexts.
8) Should I treat the point value as exact?
No. Point mode is an approximation using a chosen σ. Exact delta behavior is defined through integrals and distribution identities, not ordinary pointwise values.