Dirac_delta_function
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Definition
The Dirac delta function $\delta(x)$ is a generalized function (distribution) defined by its action on test functions. For any Schwartz_function $\varphi$:
$$ \int_{-\infty}^{\infty} \delta(x) \varphi(x) \, dx = \varphi(0) $$More generally, the delta function centered at $x_0$:
$$ \int_{-\infty}^{\infty} \delta(x - x_0) \varphi(x) \, dx = \varphi(x_0) $$This is called the sifting property.
Symbolic Properties
While not a function in the classical sense, $\delta(x)$ is often written symbolically as:
$$ \delta(x) = \begin{cases} +\infty & x = 0 \\ 0 & x \neq 0 \end{cases} $$with the normalization:
$$ \int_{-\infty}^{\infty} \delta(x) \, dx = 1 $$Delta Sequences (Nascent Deltas)
Since the delta function is not a true function, it is often represented as the limit of a sequence of ordinary functions. A common choice is the Gaussian:
$$ \delta(x) = \lim_{N \to \infty} N e^{-\pi N^2 x^2} $$This works because:
- Height = $N$ at $x = 0$
- Width $\propto 1/N$
- Area = 1 for all $N$:
As $N \to \infty$, the Gaussian becomes infinitely tall and infinitely narrow while maintaining unit area—exactly the defining behavior of $\delta(x)$.
The factor of $\pi$ in the exponent is chosen so the integral equals 1 without additional normalization.
Other common delta sequences include:
- Lorentzian: $\displaystyle\frac{1}{\pi} \frac{\epsilon}{x^2 + \epsilon^2}$ as $\epsilon \to 0$
- Sinc: $\displaystyle\frac{\sin(Nx)}{\pi x}$ as $N \to \infty$
Properties
Sifting Property: $\int f(x) \delta(x - x_0) \, dx = f(x_0)$
Scaling: $\delta(ax) = \frac{1}{|a|} \delta(x)$ for $a \neq 0$
Symmetry: $\delta(-x) = \delta(x)$
Fourier_transform: $\mathcal{F}\{\delta(x)\} = 1$ and $\mathcal{F}\{1\} = \delta(\xi)$
convolution Identity: $f(x) * \delta(x) = f(x)$
Relation to Dirac_measure
The Dirac delta function can be viewed as the density of the Dirac_measure $\delta_0$ with respect to Lebesgue measure (in a distributional sense).