Dirac delta function

The Dirac delta is the distribution defined by $⟨\delta ,\phi ⟩=\phi (0)$.

As distributions on the line, $m\sin (m|t|)\to 2{\delta }_0$. Pairing with a test function $A$ folds the integral to ${\int }_0^{\infty }m\sin (ms)[A(s)+A(-s)]ds$. One integration by parts yields $2A(0)$ plus a cosine integral tending to zero by the Riemann-Lebesgue lemma. Pointwise convergence is unnecessary.

For a smooth hypersurface $S$, its surface delta is the distribution $⟨{\delta }_S,\phi ⟩={\int }_S\phi dS$. Its normalization uses geometric surface measure rather than a defining function. On the unit circle, ${\delta }_{S^1}$ pairs by integrating $\phi (\cos t,\sin t)$ over $0\le t\le 2\pi$.

For the geometric surface delta distribution on the radius-$a$ sphere in ${\mathbb{R}}^3$, $⟨u_a,\varphi ⟩=a^2{\int }_{S^2}\varphi (a\omega )d\sigma (\omega )$. Angular integration gives the Fourier transform ${\hat{u}}_a(\xi )=4\pi a\sin (a|\xi |)/|\xi |$, with removable value $4\pi a^2$ at zero. The convolution of distributions with a compactly supported factor gives the regular distributionThis density has total mass $16{\pi }^2{a}^2{b}^2$. Values on endpoint spheres do not affect the distribution. When $a=b$, the inverse-distance singularity is locally integrable in three dimensions and is not an additional point mass. The annulus expresses the triangle inequality for the sum of two vectors of fixed lengths.

If $g$ is smooth and $\nabla g\ne 0$ on $S=\{g=0\}$, transverse local coordinates give $⟨\delta (g),\phi ⟩={\int }_S\phi /|\nabla g|dS$. Thus ${\delta }_S=|\nabla g|\delta (g)$ as distributions. In particular ${\delta }_{S^1}=2\delta (|x{|}^2-1)$. Omitting this Jacobian confuses the delta of a defining function with a geometric surface delta distribution.

The identityis interpreted distributionally: symmetric frequency cutoffs give $\sin (nx)/(\pi x)$, which converge to the Dirac delta function.

With inverse Fourier transform normalization $(2\pi {)}^{-1}$, $\int e^{is\theta }{\theta }^{j}d\theta =2\pi i^{-j}{\delta }^{(j)}(s)$ as an oscillatory integral. The factor $i^{-j}$ and the sign convention $⟨{\delta }^{\prime },g⟩=-g^{\prime }(0)$ are essential. For instance, amplitude $-i{x}_2\theta /(2\pi )$ with phase $x_1\theta$ gives $-x_2{\delta }^{\prime }(x_1)$.

For every test function $\phi$,so $H^{\prime }=\delta$ as distributions.

For a smooth function $u$, the distributional derivative of the product of $u$ with the Heaviside function is $D(Hu)=H{u}^{\prime }+u(0){\delta }_0$. Induction gives the displayed formula. It records all boundary sources, including derivatives of the Dirac delta distribution, and determines which initial derivatives must vanish in a retarded fundamental solution.

For nonzero $a$, $\delta (ax)=\delta (x)/|a|$ in the distributional sense.

The derivative of the Dirac delta is the distributional derivative defined byfor every test function $\phi$.