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For $\mathrm{Re}p>0$, put $\lambda =p^{1/3}$ using the principal cube root and $r_{1,2}=\lambda e^{\pm i\pi /3}$. The decaying Green function for ${\partial }_s^3+p$ on the real line isThe identities among the three characteristic roots show that $R_p$ and $R_p^{\prime }$ are continuous, while $R_p^{\prime \prime }$ has jump one. Its distributional derivative therefore satisfies $({\partial }_s^3+p)R_p={\delta }_0$. This is the kernel of the resolvent operator $(p+{\partial }_s^3{)}^{-1}$, obtained by the time Laplace transform of the Airy equation.

A BV space function is approximately differentiable at almost every point with respect to Lebesgue measure. Its approximate gradient is the density of the absolutely continuous part of its distributional derivative. This does not imply membership in a Sobolev space.

The distributional derivative of this indicator function is $e_1{\mathcal{H}}^1$ on the vertical segment $x_1=1/2$, so its total variation seminorm on a domain is one. This is singular with respect to planar Lebesgue measure, whereas Sobolev space weak derivatives in $W^{1,1}$ have integrable densities. Its bounded-variation space norm is $1/2+1=3/2$.

For locally integrable real $u$ with nonnegative distributional derivative $\Delta u$, its ball means increase with radius. To see this, mollify: for a smooth subharmonic function the derivative of its spherical mean is the integral of $\Delta u$ over the ball divided by the sphere area. The ball means inherit monotonicity; local $L^1$ convergence passes this to $u$. Their decreasing-radius limit exists, may be minus infinity at exceptional points, and equals $u$ almost everywhere by the Lebesgue differentiation theorem. It is upper semicontinuous as an infimum of continuous ball means and satisfies the mean-value inequality. Arbitrary changes of a representative on a null set can destroy its pointwise maximum principle.

For RNS string matter normalized by $[{\dot{X}}^{\mu }(\sigma ),X^{\nu }({\sigma }^{\prime })]=-i\pi {\eta }^{\mu \nu }\delta (\sigma -{\sigma }^{\prime })$ and $\{{\psi }_1^{\mu }(\sigma ),{\psi }_1^{\nu }({\sigma }^{\prime })\}=\pi {\eta }^{\mu \nu }\delta (\sigma -{\sigma }^{\prime })$, put $A={\partial }_{+}X$ and $J={\psi }_1\cdot A$. Then $[A^{\mu }(\sigma ),A^{\nu }({\sigma }^{\prime })]=(i\pi /2){\eta }^{\mu \nu }{\partial }_{\sigma }\delta (\sigma -{\sigma }^{\prime })$. The noncentral part of $\{J(\sigma ),J({\sigma }^{\prime })\}$ is $\pi \delta A^2+(i\pi /2){\partial }_{\sigma }\delta {\psi }_1(\sigma )\cdot {\psi }_1({\sigma }^{\prime })$. The distributional derivative identity $f(\sigma ,{\sigma }^{\prime }){\delta }^{\prime }=f(\sigma ,\sigma ){\delta }^{\prime }+({\partial }_{{\sigma }^{\prime }}f)(\sigma ,\sigma )\delta$, with a vanishing coincident Grassmann bilinear, converts this to $\pi \delta {\theta }_{++}$. Use the chiral equation of motion to identify ${\partial }_{\sigma }{\psi }_1={\partial }_{+}{\psi }_1$. Quantum normal ordering adds a central contact term proportional to ${\delta }^{\prime \prime }$, the local form of the N=1 super-Virasoro algebra central extension.

The locally integrable function $E(z)=1/(\pi z)$ on the plane is a fundamental solution for the Wirtinger derivative ${\partial }_{\bar{z}}=({\partial }_x+i{\partial }_y)/2$. Let $L=(4\pi {)}^{-1}\log (x^2+y^2)$ be the fundamental solution of the Laplace equation. Its first distributional derivatives agree with its ordinary ones: deleting a radius-$\epsilon$ disk in integration by parts gives an error $O(\epsilon |\log \epsilon |)$, tending to zero. Hence $E=4{\partial }_{z}L$ andThe analogous fundamental solution for ${\partial }_z$ is $1/(\pi \bar{z})$. The singular support of each kernel is exactly $\{0\}$: they are smooth elsewhere, and smoothness near zero would make their Wirtinger derivatives smooth there, contradicting the Dirac delta distribution identity.

Target: Distributional Cauchy-Riemann lemma. Parent: Cauchy-Riemann equations. Intended file: analysis.bigb.

For $L=(D_t+a)(D_t+b)$ with $a>b>0$, $H$ denotes the Heaviside step function. This Green function vanishes at negative time and satisfies $LG=\delta (t)$. Its positive-time part solves the homogeneous ordinary differential equation, is zero at the origin, and has right derivative one. Thus it is continuous with a unit derivative jump, which gives precisely the Dirac delta function in its second distributional derivative.

With the negative-exponential Fourier transform convention,Both poles lie in the upper half-plane. Contour integration closes above for positive time and below for negative time, giving the title expression. When the integral converges, convolution $y(t)={\int }_{-\infty }^{t}G(t-s)f(s)ds$ supplies the response with no freely added homogeneous transient growing toward negative infinity. The continuous limit $a\to b$ is $G(t)=H(t)t{e}^{-bt}$.

If $u_j\to u$ in weak convergence of distributions, then every fixed distributional derivative converges too. For each test function $\varphi$, its derivative $D^{\alpha }\varphi$ is another test function, andThus a series converging as distributions can be differentiated term by term in that topology. For example, uniform convergence on compact sets implies convergence of the associated regular distributions, even when the ordinary series of derivatives does not converge pointwise.

Target: Even-weight principal-value distribution. Parent: Principal-value reciprocal distribution. Intended file: distribution-theory.bigb.

At fixed spatial momentum with mode energy $E>0$, $g_E(t)=e^{-iE|t|}/(2E)$ is continuous, with derivatives $-i/2$ and $+i/2$ at zero from opposite sides. Hence its second distributional derivative has a $-i\delta (t)$ term, while away from zero $g_E^{\prime \prime }+E^2{g}_E=0$. Therefore $({\partial }_t^2+E^2)g_E=-i\delta (t)$ and, after the spatial Fourier transform, $(\square +m^2){\Delta }_F=-i{\delta }^4$. This checks the propagator numerator, contour signs and equal-time canonical commutation relations independently of residue calculation.

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

The locally integrable function $\log |x|$ has distributional derivative equal to the principal-value reciprocal distribution. Symmetric integration by parts makes the boundary term $\log \epsilon [\phi (\epsilon )-\phi (-\epsilon )]$ tend to zero.

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.

Let $y$ be piecewise twice continuously differentiable, with finite one-sided values and derivatives at $c$, and write $[y{]}_c=y(c+)-y(c-)$. Its distributional derivative is $Dy=y_{\mathrm{reg}}^{\prime }+[y{]}_c{\delta }_c$, and differentiating again givesHere ${\delta }_c$ is the Dirac delta at $c$, and the subscript reg means the ordinary derivative on the two open intervals, interpreted as a regular distribution. To prove the first formula, split $-\int y{\phi }^{\prime }$ at $c$ and use integration by parts on each piece for a test function $\phi$. The two boundary contributions add to $[y{]}_c\phi (c)$. Applying the same argument to the regular first derivative proves the second formula. Thus a derivative jump produces a delta source, whereas a value jump produces the derivative of a delta source.

Distribution theory extends ordinary functions by letting them act on smooth test functions. A locally integrable function $f$ acts by $⟨f,\varphi ⟩=\int f\varphi$, while the Dirac delta distribution acts by $⟨{\delta }_a,\varphi ⟩=\varphi (a)$. The latter is a point source, rather than an ordinary function with a very large value. The space of test functions and its topology make distributions continuous linear functionals.^[ref]

Integration by parts supplies a distributional derivative even when ordinary differentiability fails:For the Heaviside function $H$, $⟨H^{\prime },\varphi ⟩=-{\int }_0^{\infty }{\varphi }^{\prime }(x)dx=\varphi (0)$, so $H^{\prime }={\delta }_0$. A jump has become a point source under differentiation. This is useful for impulses in differential equations, concentrated loads and interface jump conditions.

The extension has limits: multiplying a distribution by a smooth function is always defined, but arbitrary products of distributions are not. The theory controls which operations are meaningful through their action on test functions, rather than by manipulating symbolic point values.

If a smooth functional has integrand $f(x,v)$ independent of the function value, a weak local minimizer satisfies the displayed first integral almost everywhere. Its first variation gives $\int f_v(x,u^{\prime }){\phi }^{\prime }=0$ for every compactly supported test function. Thus the distributional derivative of $f_v(x,u^{\prime })$ vanishes, making it constant on the interval. Equivalently, any smooth compactly supported function with integral zero is the derivative of such a test function, so the first-variation identity says that $f_v(x,u^{\prime })$ agrees with a single constant against all tests. This reduces the Euler-Lagrange equation from a second-order equation to an algebraic relation in $u^{\prime }$.

The period-$L$ extension of $f(x)=Kx/L$ on $-L/2<x<L/2$ has Fourier coefficients $a_0=a_n=0$ and $b_n=K(-1{)}^{n+1}/(\pi n)$. ConsequentlyThe Fourier series converges to the ramp away from its jumps and to zero at every jump, the average of the one-sided limits. For $K\ne 0$, Termwise differentiation of a Fourier series fails here as an ordinary convergent series: the differentiated summands have nondecaying amplitude $2K/L$. The classical derivative equals $K/L$ between jumps, whereas the periodic distributional derivative includes negative point masses of weight $K$ at the jumps.

The distributional derivative of $\log |x|$ is the principal-value reciprocal distribution. Its transform is $-i\pi \mathrm{sgn}k$. Dividing the derivative-transform relation by $ik$ gives the displayed finite-part inverse-absolute-value distribution, with an undetermined delta term fixed by the finite-part convention.

With the Fourier transform convention $\hat{f}(\omega )={\int }_{\mathbb{R}}f(t)e^{-i\omega t}dt$, the regular order-zero Bessel function of the first kind has the tempered distribution transformThe density is integrable despite its endpoint singularities. Fourier inversion gives the equivalent integral representationThe second formula follows by $\omega =\cos \theta$ and cancellation of the odd imaginary part. It is the standard real integral representation of $J_0$.^[ref]

Indeed, transforming the order-zero Bessel differential equation $t{y}^{\prime \prime }+y^{\prime }+ty=0$ gives $(1-{\omega }^2)Y^{\prime }-\omega Y=0$. Inside $(-1,1)$, its solutions are $C(1-{\omega }^2{)}^{-1/2}$. The compactly supported choice also solves the equation as a distribution: rewrite it as $[(1-{\omega }^2)Y{]}^{\prime }+\omega Y=0$, and observe that $(1-{\omega }^2)Y=C\sqrt{1-{\omega }^2}{1}_{(-1,1)}$ is continuous and zero at the two endpoints. Its distributional derivative has no endpoint Dirac delta terms. The inverse is regular at $t=0$, and $y(0)=1$ fixes $C=2$.

The transformed equation alone also allows separate constants on $\omega <-1$ and $\omega >1$. Selecting the compactly supported density amounts to selecting the solution regular at the origin, rather than every possible singular solution of the Bessel differential equation.

For $L=a_2(x)D^2+a_1(x)D+a_0(x)$ with $a_2(\xi )\ne 0$, a Green's function solving $L_{x}G=\delta (x-\xi )$ is homogeneous on either side of $\xi$, continuous there, and has first derivative jump $[G_x{]}_{-}^{+}=1/a_2(\xi )$. A nonzero jump in $G$ itself would produce an unwanted distributional derivative of the Dirac delta. Once continuity removes that term, integrating the equation across $\xi$ fixes the derivative jump. Endpoint boundary conditions complete the classical characterization.

A Hadamard finite-part integral removes explicit divergent terms from a truncated integral and retains the finite limit. For example, the one-sided finite part of $H(x)/x$ can be defined byfor each test function $\varphi$. The logarithmic subtraction cancels the divergence at zero. The locally integrable function $L(x)=H(x)\log x$, interpreted as zero for $x<0$, has exactly this distributional derivative: integrate $-{\int }_0^{\infty }\log x{\varphi }^{\prime }(x)dx$ by parts with lower cutoff $ϵ$.

This is distinct from the symmetric principal-value reciprocal distribution, which is the derivative of $\log |x|$ on the whole line. More singular reciprocal powers require additional Taylor polynomial subtractions. The subtraction and normalization must be specified, because distributions supported at the singular point can distinguish different extensions.