DLMF:1.16.E27 (Q627)

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DLMF:1.16.E27
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    Statements

    Defining formula
    ⟨ D k ⁢ f , ϕ ⟩ = ( - 1 ) | k | ⁢ ∫ ℝ n f ⁢ ( x ) ⁢ ϕ ( k ) ⁢ ( x ) ⁢ d x , distribution superscript 𝐷 𝑘 𝑓 italic-ϕ superscript 1 𝑘 subscript 𝑛 𝑓 𝑥 superscript italic-ϕ 𝑘 𝑥 𝑥 {\displaystyle{\displaystyle\left\langle D^{k}f,\phi\right\rangle=(-1)^{\left|% k\right|}\int_{{\mathbb{R}^{n}}}f(x)\phi^{(k)}(x)\mathrm{d}x,}}
    0 references
    DLMF id
    DLMF:1.16.E27
    0 references
    constraint
    ϕ ∈ 𝒟 n italic-ϕ subscript 𝒟 𝑛 {\displaystyle{\displaystyle\phi\in\mathcal{D}_{n}}}
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    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    Q10905
    Defining formula
    ⟨ Λ , ϕ ⟩ distribution Λ italic-ϕ {\displaystyle{\displaystyle\left\langle\NVar{\Lambda},\NVar{\phi}\right% \rangle}}
    xml-id
    C1.S16.SS1.p5.m5aldec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2aedec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    Q10812
    Defining formula
    ℝ {\displaystyle{\displaystyle\mathbb{R}}}
    xml-id
    introduction.Sx4.p2.t1.r14.m2aadec
    0 references
    Q10756
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C1.S1.XMD4.m1bdec
    0 references
    Q10757
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C1.S1.XMD7.m1idec
    0 references
    Q10911
    Defining formula
    𝒟 n subscript 𝒟 𝑛 {\displaystyle{\displaystyle\mathcal{D}_{n}}}
    xml-id
    C1.S16.XMD10.m1adec
    0 references
    Q10912
    Defining formula
    ϕ ⁢ ( x 1 , x 2 , … , x n ) italic-ϕ subscript 𝑥 1 subscript 𝑥 2 … subscript 𝑥 𝑛 {\displaystyle{\displaystyle\phi(x_{1},x_{2},\dots,x_{n})}}
    xml-id
    C1.S16.XMD13.m1adec
    0 references
    Q10909
    Defining formula
    D ⁢ f 𝐷 𝑓 {\displaystyle{\displaystyle Df}}
    xml-id
    C1.S16.XMD6.m1fdec
    0 references
     
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