Dirac Delta Calculator Inputs
Use the fields below to evaluate impulse approximations, test the sifting property, and inspect the scaling identity for shifted delta terms.
Example Data Table
This example uses a = 0 and ε = 0.5 to illustrate how impulse approximations behave near the shift point.
| x | Gaussian | Rectangular | Sinc |
|---|---|---|---|
| -1.0 | 0.020667 | 0.000000 | 0.289440 |
| -0.5 | 0.415107 | 1.000000 | 0.636620 |
| 0.0 | 1.128379 | 1.000000 | 0.636620 |
| 0.5 | 0.415107 | 1.000000 | 0.636620 |
| 1.0 | 0.020667 | 0.000000 | 0.289440 |
Formula Used
1) Gaussian Approximation
δε(x-a) = exp(-((x-a)/ε)2) / (√π · ε)
2) Rectangular Approximation
δε(x-a) = 1 / (2ε), when |x-a| ≤ ε; otherwise 0
3) Sinc Approximation
δε(x-a) = sin((x-a)/ε) / (π(x-a))
4) Sifting Property
∫ k·f(x)·δ(x-a) dx = k·f(a)
5) Scaling Identity
δ(bx-c) = (1/|b|) · δ(x-c/b), for b ≠ 0
How to Use This Calculator
- Enter the evaluation point x where you want the approximation value.
- Enter the impulse location a to shift the delta center.
- Choose a positive ε value to control peak width.
- Enter f(a) and multiplier k to test the sifting property.
- Enter b and c to inspect the scaling identity δ(bx-c).
- Set graph limits and sample points for the plot.
- Press the calculate button to display results above the form.
- Use the CSV and PDF buttons to export the calculated output.
Frequently Asked Questions
1) What does the Dirac delta represent?
It represents an idealized impulse concentrated at one point. Its total area equals one, while its pointwise height is not treated like an ordinary function value.
2) Why does this calculator use approximations?
Because the Dirac delta is a distribution, numerical work uses narrow families such as Gaussian, rectangular, or sinc approximations that mimic its defining properties.
3) What is the role of ε?
ε controls concentration width. Smaller ε creates a narrower, taller peak and generally improves the visual resemblance to an impulse when the numerical range is large enough.
4) What does the sifting result mean?
The sifting property extracts the value of a test function at the impulse location. That is why the integral reduces directly to k·f(a).
5) Why can the area be close to one?
Approximations are normalized to integrate to one over a sufficiently wide interval. A smaller graph range may cut off the tails and slightly change the computed area.
6) What does δ(bx-c) show?
It shifts the impulse to x = c/b and scales the strength by 1/|b|. Both effects are essential when changing variables inside delta expressions.
7) Which approximation should I use?
Gaussian is smooth and stable. Rectangular is intuitive and simple. Sinc is useful in signal contexts, though it oscillates and may need wider plotting ranges.
8) Can this calculator replace symbolic distribution theory?
No. It is best for learning, numerical intuition, plotting, and checking common identities. Rigorous proofs still rely on distribution theory and test functions.