Calculator Input
Example Data Table
| Case | f(x) | Distribution | Interval | Rule | Result |
|---|---|---|---|---|---|
| Shifted impulse | x^2 + 1 | δ(x - 2) | [0, 5] | f(2) | 5 |
| Scaled argument | sin(x) | δ(3*x - 6) | [0, 4] | sin(2) / 3 | 0.3031 |
| Outside interval | exp(-x^2) | δ(x - 7) | [-2, 2] | No contribution | 0 |
| First derivative | x^3 | δ′(x - 2) | [0, 4] | -f′(2) | -12 |
Formula Used
Shifted impulse:
∫ f(x)δ(x - c) dx = f(c), when c is inside the integration interval.
Linear argument:
δ(a*x + b) = δ(x - r) / |a|, where r = -b/a.
∫ f(x)δ(a*x + b) dx = f(-b/a) / |a|.
First derivative impulse:
∫ f(x)δ′(x - c) dx = -f′(c).
Gaussian learning approximation:
δε(x) = exp(-x² / ε²) / (ε√π). It is used only for graphing and numerical comparison.
How to Use This Calculator
- Enter the test function in terms of
x. - Select the distribution type.
- Enter bounds for the integral.
- For
δ(a*x + b), enter values foraandb. - For
δ(x - c)orδ′(x - c), enterc. - Choose an endpoint convention if the impulse is exactly on a bound.
- Press calculate to see the result above the form.
- Use CSV or PDF buttons to save the calculation.
Dirac Delta Generalized Function Guide
Meaning
The Dirac delta is not an ordinary function. It is a generalized function. It represents a unit impulse placed at one point. Its value is handled through integrals, not through normal point evaluation.
Sifting Property
The main rule is the sifting property. When a smooth function is multiplied by δ(x − c), the integral selects f(c). This is useful in signals, physics, probability, and differential equations. The calculator checks whether the impulse point is inside the chosen interval. If it is outside, the integral contribution is zero.
Scaled Arguments
Scaled arguments need extra care. For δ(a x + b), the root is x = −b / a. The impulse weight becomes 1 / |a|. The integral equals f(root) / |a| when the root lies within the bounds. This makes changes of variables consistent.
Derivative Impulses
Derivative impulses behave differently. The first derivative δ′(x − c) returns the negative derivative of the test function. So the integral of f(x)δ′(x − c) equals −f′(c). This calculator estimates f′(c) by a centered numerical difference. A smaller step can improve precision, but extremely tiny steps may cause rounding errors.
Approximation View
The Gaussian and rectangular views are only approximations. They help students see a narrow spike. They are not the actual distribution. A true delta has zero width and unit area. The plotted curve therefore supports learning, while the result uses distribution rules.
Input Tips
Use clean expressions for f(x). Examples include x^2 + 3*x, sin(x), exp(-x^2), and sqrt(x+4). Choose bounds that match the problem. Set a and b for a linear argument, or enter c for shifted forms. Review the root, weight, and final integral before exporting.
Advanced Note
For advanced work, remember that distribution formulas assume smooth test functions near the impulse point. Discontinuous inputs need special interpretation. Endpoint impulses may use full or half weight, depending on convention. This page lets you choose that rule, which makes homework, engineering notes, and symbolic checks easier to document with fewer manual mistakes during reviews.
Best Use
This tool is best for study and checking work. It explains each step and creates downloadable records. It also shows example cases, so users can compare their answer with known delta identities.
FAQs
1. Is the Dirac delta a normal function?
No. It is a generalized function, also called a distribution. It is defined by how it acts inside integrals.
2. What does the sifting property mean?
It means the integral selects the value of the test function at the impulse location, when that point lies inside the bounds.
3. Why does δ(a*x + b) need division by |a|?
The factor appears from variable scaling. A steeper argument compresses the impulse, so the area must be adjusted by 1 / |a|.
4. What happens if the impulse is outside the interval?
The contribution is zero. The delta only samples the function when its root lies within the selected integration bounds.
5. What is δ′(x - c)?
It is the first distribution derivative of the shifted impulse. In an integral, it returns the negative derivative of the test function at c.
6. Is the plotted spike the true delta?
No. The graph uses a Gaussian approximation. The real delta has no ordinary height or width, but it has unit area.
7. Which functions can I enter?
You can use x, numbers, pi, e, powers, and common functions like sin, cos, tan, exp, log, sqrt, and abs.
8. Why is endpoint handling optional?
Different courses and applications use different endpoint conventions. The calculator allows full, half, or zero endpoint contribution for clarity.