Revision #1159 → #1814 · back to history
modifiedDirac delta function6ea46e9750b4
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| anchors | [{"section":"(Lead)","snippet":"the Dirac delta function (or"},{"type":"math_alttext","value":"{\\displaystyle \\delta (x)={\\begin{cases}0,&x\\neq 0\\\\{\\infty },&x=0\\end{cases}}}"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\infty }^{\\infty }\\delta (x)\\,dx=1.}"}] | — |
| note | The Dirac delta is formalized both as `Measure.dirac`, `Distribution.delta`, and `TemperedDistribution.delta`. | The Dirac delta is formalized as `MeasureTheory.Measure.dirac`, `Distribution.delta`, and `TemperedDistribution.delta`. |
modifiedHeuristic characterization4330a51cf1b4
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| anchors | [{"section":"Definitions","snippet":"can be loosely thought of as a function on the real line which is zero everywhere except at the origin"},{"type":"math_alttext","value":"{\\displaystyle \\delta (x)\\simeq {\\begin{cases}+\\infty ,&x=0\\\\0,&x\\neq 0\\end{cases}}}"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\infty }^{\\infty }\\delta (x)\\,dx=1.}"}] | — |
| note | This is an informal heuristic description of δ and has no formalized counterpart. | This is an informal heuristic description of δ with no formalized counterpart. |
modifiedDirac measure1db8f5df7276
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| anchors | [{"section":"As a measure","snippet":"to define a measure , called Dirac measure"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\infty }^{\\infty }f(x)\\,\\delta (dx)=f(0)}"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\infty }^{\\infty }f(x)\\,\\delta (x)\\,dx=f(0)}"}] | — |
| note | Defined as `MeasureTheory.Measure.dirac` with `dirac_apply_of_mem` giving the value 1 on sets containing the point. | Defined as `MeasureTheory.Measure.dirac` in `Mathlib.MeasureTheory.Measure.Dirac`. |
modifiedDelta measure is singularb50542504268
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| anchors | [{"section":"As a measure","snippet":"The measure [MATH] is not absolutely continuous with respect to the Lebesgue measure"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\infty }^{\\infty }f(x)\\,\\delta (dx)=f(0)}"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\infty }^{\\infty }f(x)\\,\\delta (x)\\,dx=f(0)}"}] | — |
| note | `mutuallySingular_dirac` shows `dirac x ⟂ₘ μ` for any `NoAtoms μ`, which specializes to Lebesgue and implies non-absolute-continuity. | `mutuallySingular_dirac` shows `dirac x ⟂ₘ μ` for any `NoAtoms μ`, specializing to Lebesgue. |
modifiedCDF of delta measure is unit step2069fb6f0c0c
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| anchors | [{"section":"As a measure","snippet":"the delta measure is characterized by its cumulative distribution function , which is the unit step function"},{"type":"math_alttext","value":"{\\displaystyle H(x)={\\begin{cases}1&{\\text{if }}x\\geq 0\\\\0&{\\text{if }}x<0.\\end{cases}}}"},{"type":"math_alttext","value":"{\\displaystyle H(x)=\\int _{\\mathbf {R} }\\mathbf {1} _{(-\\infty ,x]}(t)\\,\\delta (dt)=\\delta \\!\\left((-\\infty ,x]\\right),}"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\infty }^{\\infty }f(x)\\,\\delta (dx)=\\int _{-\\infty }^{\\infty }f(x)\\,dH(x).}"}] | — |
modifiedDelta distribution93ed67ec8573
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| note | `Distribution.delta` is the linear functional `f ↦ f x` on test functions in `𝓓^{n}(Ω, ℝ)`. | `Distribution.delta` is the linear functional `f ↦ f x` on test functions. |
modifiedContinuity criterion for distributions21626044f794
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| anchors | [{"section":"As a distribution","snippet":"it is necessary and sufficient that, for every positive integer"},{"type":"math_alttext","value":"{\\displaystyle \\left|S[\\varphi ]\\right|\\leq C_{N}\\sum _{k=0}^{M_{N}}\\sup _{x\\in [-N,N]}\\left|\\varphi ^{(k)}(x)\\right|}"}] | — |
| note | Mathlib characterizes continuity on 𝓓 via the LF inductive-limit structure, but the explicit seminorm/C·sup-of-derivatives form stated in the article is not isolated. | Mathlib characterizes continuity on 𝓓 via the LF inductive-limit structure, not the explicit seminorm/C·sup-of-derivatives form. |
modifiedDelta as distributional derivative of Heavisideacf2a662c714
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| anchors | [{"section":"As a distribution","snippet":"it is the distributional derivative of the Heaviside step function"},{"type":"math_alttext","value":"{\\displaystyle \\delta [\\varphi ]=-\\int _{-\\infty }^{\\infty }\\varphi '(x)\\,H(x)\\,dx.}"}] | — |
modifiedDirac delta on locally compact Hausdorff space47d39c380756
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| note | `Measure.dirac` provides the Radon measure δ_a on any measurable space, including locally compact Hausdorff spaces. | `Measure.dirac` provides δ_a on any measurable space, including locally compact Hausdorff spaces. |
modifiedScaling propertydbab358e4a42
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| anchors | [{"section":"Scaling and symmetry","snippet":"The delta function satisfies the following scaling property for a non-zero scalar"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\infty }^{\\infty }\\delta (\\alpha x)f(x)\\,dx=\\int _{-\\infty }^{\\infty }\\delta (u)f(u/\\alpha )\\,{\\frac {du}{|\\alpha |}}={\\frac {f(0)}{|\\alpha |}}}"}] | — |
modifiedIndefinite integral of deltab227b78ac000
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| anchors | [{"section":"Indefinite integral","snippet":"For a constant [MATH] and a \"well-behaved\" arbitrary real-valued function"},{"type":"math_alttext","value":"{\\displaystyle \\displaystyle {\\int }y(x)\\delta (x-a)dx=y(a)H(x-a)+c,}"}] | — |
modifiedScaling in n dimensions60d5b1824763
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| anchors | [{"section":"Properties in n dimensions","snippet":"The delta distribution in an n -dimensional space satisfies the following scaling property"},{"type":"math_alttext","value":"{\\displaystyle \\delta (\\alpha {\\boldsymbol {x}})=|\\alpha |^{-n}\\delta ({\\boldsymbol {x}})~,}"}] | — |
modifiedInvariance under reflection/rotationd051359801cd
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| anchors | [{"section":"Properties in n dimensions","snippet":"Under any reflection or rotation ρ , the delta function is invariant"},{"type":"math_alttext","value":"{\\displaystyle \\delta (\\rho {\\boldsymbol {x}})=\\delta ({\\boldsymbol {x}})~.}"}] | — |
modifiedComposition with bi-Lipschitz function4a7c9ca4cb10
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| anchors | [{"section":"Properties in n dimensions","snippet":"it is possible to define the composition of δ with a bi-Lipschitz function"},{"type":"math_alttext","value":"{\\displaystyle \\int _{\\mathbb {R} ^{n}}\\delta (g({\\boldsymbol {x}}))\\,f(g({\\boldsymbol {x}}))\\left|\\det g'({\\boldsymbol {x}})\\right|d{\\boldsymbol {x}}=\\int _{g(\\mathbb {R} ^{n})}\\delta ({\\boldsymbol {u}})f({\\boldsymbol {u}})\\,d{\\boldsymbol {u}}}"}] | — |
modifiedCoarea/submersion composition8557cd16abce
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| anchors | [{"section":"Properties in n dimensions","snippet":"Using the coarea formula from geometric measure theory"},{"type":"math_alttext","value":"{\\displaystyle \\int _{\\mathbb {R} ^{n}}f({\\boldsymbol {x}})\\,\\delta (g({\\boldsymbol {x}}))\\,d{\\boldsymbol {x}}=\\int _{g^{-1}(0)}{\\frac {f({\\boldsymbol {x}})}{|{\\boldsymbol {\\nabla }}g|}}\\,d\\sigma ({\\boldsymbol {x}})}"}] | — |
modifiedSimple layer distribution on hypersurfacef8d7f8838dc7
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| anchors | [{"section":"Properties in n dimensions","snippet":"if S is a smooth hypersurface of R n , then we can associate to S the distribution"},{"type":"math_alttext","value":"{\\displaystyle \\delta _{S}[g]=\\int _{S}g({\\boldsymbol {s}})\\,d\\sigma ({\\boldsymbol {s}})}"}] | — |
modifiedDerivative of the delta distribution761a3f630657
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| anchors | [{"section":"Derivatives","snippet":"The derivative of the Dirac delta distribution, denoted δ′ and also called the Dirac delta prime"},{"type":"math_alttext","value":"{\\displaystyle \\delta '[\\varphi ]=-\\delta [\\varphi ']=-\\varphi '(0).}"}] | — |
| note | Mathlib has `derivCLM` for tempered distributions, so `derivCLM (delta x)` is δ′, but no dedicated `delta'` decl with its API exists. | Mathlib has `derivCLM` for tempered distributions, so `derivCLM (delta x)` is δ′, but no dedicated `delta'` decl exists. |
modifiedDelta is infinitely differentiable7a6172e774aa
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| note | All tempered distributions admit `derivCLM` of every order, so δ is implicitly infinitely differentiable, but no explicit statement is given. | All tempered distributions admit `derivCLM` of every order, so δ is implicitly infinitely differentiable, but no explicit statement. |
modifiedDerivative as limit of difference quotientsf3374730a83e
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| anchors | [{"section":"Derivatives","snippet":"The first derivative of the delta function is the distributional limit of the difference quotients"},{"type":"math_alttext","value":"{\\displaystyle \\delta '(x)=\\lim _{h\\to 0}{\\frac {\\delta (x+h)-\\delta (x)}{h}}.}"}] | — |
modifiedBasic properties of delta derivative22a77064b950
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| anchors | [{"section":"Derivatives","snippet":"The derivative of the delta function satisfies a number of basic properties"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\delta '(-x)&=-\\delta '(x)\\\\x\\delta '(x)&=-\\delta (x)\\end{aligned}}}"}] | — |
modifiedDelta distribution on open set in higher dimensions08009b02df76
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| anchors | [{"section":"Higher dimensions","snippet":"the Dirac delta distribution centered at a point a ∈ U is defined"},{"type":"math_alttext","value":"{\\displaystyle \\delta _{a}[\\varphi ]=\\varphi (a)}"}] | — |
modifiedAlpha derivative of delta_a7e5045724311
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| anchors | [{"section":"Higher dimensions","snippet":"denotes the associated mixed partial derivative operator, then the α -th derivative"},{"type":"math_alttext","value":"{\\displaystyle \\delta _{a}[\\varphi ]=\\varphi (a)}"}] | — |
modifiedStructure of distributions with point support36a54f6ad5e4
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| anchors | [{"section":"Higher dimensions","snippet":"Higher derivatives enter into mathematics naturally as the building blocks for the complete structure of distributions with point support"},{"type":"math_alttext","value":"{\\displaystyle S=\\sum _{|\\alpha |\\leq m}c_{\\alpha }\\partial ^{\\alpha }\\delta _{a}.}"}] | — |
modifiedApproximate delta function / approximation to the identity302b6560a0b6
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| anchors | [{"section":"Approximations to the identity","snippet":"An approximate delta function η ε can be constructed in the following manner"},{"type":"math_alttext","value":"{\\displaystyle \\eta _{\\varepsilon }(x)=\\varepsilon ^{-1}\\eta \\left({\\frac {x}{\\varepsilon }}\\right).}"}] | — |
| note | `ContDiffBump.normed` provides normalized bumps used as approximations to the identity, but `η_ε` as a general rescaling construction is not isolated. | `ContDiffBump.normed` provides normalized bumps used as approximations to the identity, but η_ε as a general rescaling construction is not isolated. |
modifiedMollifier09504fff738b
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| note | `ContDiffBump` provides smooth, compactly supported bumps that are mollifiers; the term `Mollifier` itself is not introduced as a separate definition. | `ContDiffBump` provides smooth, compactly supported bumps that are mollifiers; no dedicated `Mollifier` name is introduced. |
modifiedRectangular function approximationcab3aa079310
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| anchors | [{"section":"Probabilistic considerations","snippet":"if η 1 is the uniform distribution"},{"type":"math_alttext","value":"{\\displaystyle \\eta _{\\varepsilon }(x)={\\frac {1}{\\varepsilon }}\\operatorname {rect} \\left({\\frac {x}{\\varepsilon }}\\right)={\\begin{cases}{\\frac {1}{\\varepsilon }},&-{\\frac {\\varepsilon }{2}}<x<{\\frac {\\varepsilon }{2}},\\\\0,&{\\text{otherwise}}.\\end{cases}}}"}] | — |
modifiedWigner semicircle approximation1d2ace2fffca
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| anchors | [{"section":"Probabilistic considerations","snippet":"Another example is with the Wigner semicircle distribution"},{"type":"math_alttext","value":"{\\displaystyle \\eta _{\\varepsilon }(x)={\\begin{cases}{\\frac {2}{\\pi \\varepsilon ^{2}}}{\\sqrt {\\varepsilon ^{2}-x^{2}}},&-\\varepsilon <x<\\varepsilon ,\\\\0,&{\\text{otherwise}}.\\end{cases}}}"}] | — |
modifiedConvolution semigroup9732cbe77ec1
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| anchors | [{"section":"Semigroups","snippet":"Approximations to the delta functions often arise as convolution semigroups"},{"type":"math_alttext","value":"{\\displaystyle \\eta _{\\varepsilon }*\\eta _{\\delta }=\\eta _{\\varepsilon +\\delta }}"}] | — |
modifiedHeat kernel40a2f66107b4
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| anchors | [{"section":"The heat kernel","snippet":"The heat kernel , defined by"},{"type":"math_alttext","value":"{\\displaystyle \\eta _{\\varepsilon }(x)={\\frac {1}{\\sqrt {2\\pi \\varepsilon }}}\\mathrm {e} ^{-{\\frac {x^{2}}{2\\varepsilon }}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {\\partial u}{\\partial t}}={\\frac {1}{2}}{\\frac {\\partial ^{2}u}{\\partial x^{2}}}.}"}] | — |
modifiedHeat kernel in higher dimensions26841d5bc8db
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| anchors | [{"section":"The heat kernel","snippet":"In higher-dimensional Euclidean space R n , the heat kernel is"},{"type":"math_alttext","value":"{\\displaystyle \\eta _{\\varepsilon }={\\frac {1}{(2\\pi \\varepsilon )^{n/2}}}\\mathrm {e} ^{-{\\frac {x\\cdot x}{2\\varepsilon }}},}"}] | — |
modifiedPoisson kernel982a161ea77e
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| anchors | [{"section":"The Poisson kernel","snippet":"The Poisson kernel"},{"type":"math_alttext","value":"{\\displaystyle \\eta _{\\varepsilon }(x)={\\frac {1}{\\pi }}\\mathrm {Im} \\left\\{{\\frac {1}{x-\\mathrm {i} \\varepsilon }}\\right\\}={\\frac {1}{\\pi }}{\\frac {\\varepsilon }{\\varepsilon ^{2}+x^{2}}}={\\frac {1}{2\\pi }}\\int _{-\\infty }^{\\infty }\\mathrm {e} ^{\\mathrm {i} \\xi x-|\\varepsilon \\xi |}\\,d\\xi }"}] | — |
| mathlib.module | Mathlib.Analysis.Complex.Harmonic.Poisson | Mathlib.Analysis.Complex.Poisson |
modifiedAiry function approximation928745c25f36
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| anchors | [{"section":"Oscillatory integrals","snippet":"is the rescaled Airy function"},{"type":"math_alttext","value":"{\\displaystyle \\varepsilon ^{-1/3}\\operatorname {Ai} \\left(x\\varepsilon ^{-1/3}\\right).}"}] | — |
modifiedWave equation Cauchy problemd3ec52d0cc6d
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| anchors | [{"section":"Oscillatory integrals","snippet":"Another example is the Cauchy problem for the wave equation"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}c^{-2}{\\frac {\\partial ^{2}u}{\\partial t^{2}}}-\\Delta u&=0\\\\u=0,\\quad {\\frac {\\partial u}{\\partial t}}=\\delta &\\qquad {\\text{for }}t=0.\\end{aligned}}}"}] | — |
modifiedSinc function approximation9cce19f94891
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| anchors | [{"section":"Oscillatory integrals","snippet":"Other approximations to the identity of this kind include the sinc function"},{"type":"math_alttext","value":"{\\displaystyle \\eta _{\\varepsilon }(x)={\\frac {1}{\\pi x}}\\sin \\left({\\frac {x}{\\varepsilon }}\\right)={\\frac {1}{2\\pi }}\\int _{-{\\frac {1}{\\varepsilon }}}^{\\frac {1}{\\varepsilon }}\\cos(kx)\\,dk}"}] | — |
modifiedBessel function approximation490deeb7b37b
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| anchors | [{"section":"Oscillatory integrals","snippet":"and the Bessel function"},{"type":"math_alttext","value":"{\\displaystyle \\eta _{\\varepsilon }(x)={\\frac {1}{\\varepsilon }}J_{\\frac {1}{\\varepsilon }}\\left({\\frac {x+1}{\\varepsilon }}\\right).}"}] | — |
modifiedPlane wave decomposition of deltab608eda92657
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| anchors | [{"section":"Plane wave decomposition","snippet":"Then δ is obtained by applying a power of the Laplacian to the integral with respect to the unit sphere measure"},{"type":"math_alttext","value":"{\\displaystyle \\delta (x)=\\Delta _{x}^{(n+k)/2}\\int _{S^{n-1}}g(x\\cdot \\xi )\\,d\\omega _{\\xi }.}"}] | — |
modifiedInverse Fourier transform of 1a9fa1c4e786a
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| anchors | [{"section":"Fourier transform","snippet":"The inverse Fourier transform of the tempered distribution f ( ξ ) = 1 is the delta function"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\infty }^{\\infty }1\\cdot e^{2\\pi ix\\xi }\\,d\\xi =\\delta (x)}"},{"type":"math_alttext","value":"{\\displaystyle \\langle 1,{\\widehat {f}}\\rangle =f(0)=\\langle \\delta ,f\\rangle }"}] | — |
modifiedLaplace transform of delta9812f99470e3
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| anchors | [{"section":"Fourier transform","snippet":"the Laplace transform of the delta function is found to be"},{"type":"math_alttext","value":"{\\displaystyle \\int _{0}^{\\infty }\\delta (t-a)\\,e^{-st}\\,dt=e^{-sa}.}"}] | — |
modifiedDirichlet kernel89afb610ece3
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| anchors | [{"section":"Fourier kernels","snippet":"is defined by convolution (on the interval [−π,π] ) with the Dirichlet kernel"},{"type":"math_alttext","value":"{\\displaystyle D_{N}(x)=\\sum _{n=-N}^{N}e^{inx}={\\frac {\\sin \\left(\\left(N+{\\frac {1}{2}}\\right)x\\right)}{\\sin(x/2)}}.}"},{"type":"math_alttext","value":"{\\displaystyle s_{N}(f)(x)=D_{N}*f(x)=\\sum _{n=-N}^{N}a_{n}e^{inx}}"},{"type":"math_alttext","value":"{\\displaystyle a_{n}={\\frac {1}{2\\pi }}\\int _{-\\pi }^{\\pi }f(y)e^{-iny}\\,dy.}"},{"type":"math_alttext","value":"{\\displaystyle s_{N}(f)(0)=\\int _{-\\pi }^{\\pi }D_{N}(x)f(x)\\,dx\\to 2\\pi f(0)}"},{"type":"math_alttext","value":"{\\displaystyle \\delta (x)={\\frac {1}{2\\pi }}\\sum _{n=-\\infty }^{\\infty }e^{inx}}"}] | — |
modifiedDirichlet kernel tends to delta20c73ae88bad
| Field | From #1159 | To #1814 |
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| anchors | [{"section":"Fourier kernels","snippet":"A fundamental result of elementary Fourier series states that the Dirichlet kernel restricted to the interval"},{"type":"math_alttext","value":"{\\displaystyle D_{N}(x)=\\sum _{n=-N}^{N}e^{inx}={\\frac {\\sin \\left(\\left(N+{\\frac {1}{2}}\\right)x\\right)}{\\sin(x/2)}}.}"},{"type":"math_alttext","value":"{\\displaystyle s_{N}(f)(x)=D_{N}*f(x)=\\sum _{n=-N}^{N}a_{n}e^{inx}}"},{"type":"math_alttext","value":"{\\displaystyle a_{n}={\\frac {1}{2\\pi }}\\int _{-\\pi }^{\\pi }f(y)e^{-iny}\\,dy.}"},{"type":"math_alttext","value":"{\\displaystyle s_{N}(f)(0)=\\int _{-\\pi }^{\\pi }D_{N}(x)f(x)\\,dx\\to 2\\pi f(0)}"},{"type":"math_alttext","value":"{\\displaystyle \\delta (x)={\\frac {1}{2\\pi }}\\sum _{n=-\\infty }^{\\infty }e^{inx}}"}] | — |
modifiedFejér kernel185cfd0a18b1
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| anchors | [{"section":"Fourier kernels","snippet":"The method of Cesàro summation leads to the Fejér kernel"},{"type":"math_alttext","value":"{\\displaystyle F_{N}(x)={\\frac {1}{N}}\\sum _{n=0}^{N-1}D_{n}(x)={\\frac {1}{N}}\\left({\\frac {\\sin {\\frac {Nx}{2}}}{\\sin {\\frac {x}{2}}}}\\right)^{2}.}"}] | — |
modifiedFejér kernel convergence0047589a8713
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|---|
| anchors | [{"section":"Fourier kernels","snippet":"The Fejér kernels tend to the delta function in a stronger sense"},{"type":"math_alttext","value":"{\\displaystyle \\int _{-\\pi }^{\\pi }F_{N}(x)f(x)\\,dx\\to 2\\pi f(0)}"}] | — |
modifiedBergman kernel as deltaacc06b52be11
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|---|
| anchors | [{"section":"Spaces of holomorphic functions","snippet":"is represented by integration against a kernel [MATH] , the Bergman kernel"},{"type":"math_alttext","value":"{\\displaystyle \\delta _{w}[f]=f(w)={\\frac {1}{\\pi }}\\iint _{|z|<1}{\\frac {f(z)\\,dx\\,dy}{(1-{\\bar {z}}w)^{2}}}.}"}] | — |
modifiedResolution of the identity43b810213d0e
| Field | From #1159 | To #1814 |
|---|
| anchors | [{"section":"Resolutions of the identity","snippet":"is called a resolution of the identity"},{"type":"math_alttext","value":"{\\displaystyle I=\\sum _{n=1}^{\\infty }\\varphi _{n}\\varphi _{n}^{\\dagger },}"},{"type":"math_alttext","value":"{\\displaystyle \\varphi _{n}\\varphi _{n}^{\\dagger },}"}] | — |
modifiedDirac comb29a846674070
| Field | From #1159 | To #1814 |
|---|
| anchors | [{"section":"Dirac comb","snippet":"A so-called uniform \"pulse train\" of Dirac delta measures, which is known as a Dirac comb"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {\\text{Ш}} (x)=\\sum _{n=-\\infty }^{\\infty }\\delta (x-n),}"}] | — |
modifiedPoisson summation formula2cc7bc64bd0d
| Field | From #1159 | To #1814 |
|---|
| anchors | [{"section":"Dirac comb","snippet":"is precisely the Poisson summation formula"},{"type":"math_alttext","value":"{\\displaystyle (f*\\operatorname {\\text{Ш}} )(x)=\\sum _{n=-\\infty }^{\\infty }f(x-n).}"},{"type":"math_alttext","value":"{\\displaystyle (f*\\operatorname {\\text{Ш}} )^{\\wedge }={\\widehat {f}}{\\widehat {\\operatorname {\\text{Ш}} }}={\\widehat {f}}\\operatorname {\\text{Ш}} }"}] | — |
modifiedLocal time of stochastic process1cf8c346b267
| Field | From #1159 | To #1814 |
|---|
| anchors | [{"section":"Probability theory","snippet":"The local time of a stochastic process B ( t ) is given by"},{"type":"math_alttext","value":"{\\displaystyle \\ell (x,t)=\\int _{0}^{t}\\delta (x-B(s))\\,ds}"},{"type":"math_alttext","value":"{\\displaystyle \\ell (x,t)=\\lim _{\\varepsilon \\to 0^{+}}{\\frac {1}{2\\varepsilon }}\\int _{0}^{t}\\mathbf {1} _{[x-\\varepsilon ,x+\\varepsilon ]}(B(s))\\,ds}"}] | — |
modifiedMass-spring impulse equationfbe33b9bc1e9
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|---|
| anchors | [{"section":"Structural mechanics","snippet":"The governing equation of a simple mass–spring system excited by a sudden force impulse"},{"type":"math_alttext","value":"{\\displaystyle m{\\frac {d^{2}\\xi }{dt^{2}}}+k\\xi =I\\delta (t),}"}] | — |
addedRiemann–Stieltjes integral against delta2ef0abfd6f89
addedDelta as distribution of order zero with compact supportbcf5f4f406a1
addedHigher powers identity x^n δ = 075dfd2cf9099
addedDelta(x^2 − α^2) decomposition73a29a8c7523