DLMF:1.17.E7 (Q666)

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DLMF:1.17.E7
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    Statements

    Defining formula
    lim n → ∞ ⁡ n π ⁢ ∫ - ∞ ∞ e - n ⁢ ( x - a ) 2 ⁢ ϕ ⁢ ( x ) ⁢ d x = 1 2 ⁢ ϕ ⁢ ( a - ) + 1 2 ⁢ ϕ ⁢ ( a + ) . subscript → 𝑛 𝑛 𝜋 superscript subscript superscript 𝑒 𝑛 superscript 𝑥 𝑎 2 italic-ϕ 𝑥 𝑥 1 2 italic-ϕ limit-from 𝑎 1 2 italic-ϕ limit-from 𝑎 {\displaystyle{\displaystyle\lim_{n\to\infty}\sqrt{\frac{n}{\pi}}\int_{-\infty% }^{\infty}e^{-n(x-a)^{2}}\phi(x)\mathrm{d}x=\tfrac{1}{2}\phi(a-)+\tfrac{1}{2}% \phi(a+).}}
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    DLMF id
    DLMF:1.17.E7
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
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