DLMF:1.17.E7 (Q666): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
 
(2 intermediate revisions by the same user not shown)
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

{\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3acdec
Property / Symbols used
 
Property / Symbols used: Q10757 / rank
 
Normal rank
Property / Symbols used: Q10757 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q10757 / qualifier
 
xml-id: C1.S1.XMD7.m1ddec
Property / Symbols used
 
Property / Symbols used: Q10916 / rank
 
Normal rank
Property / Symbols used: Q10916 / qualifier
 
Defining formula:

ϕ ( x ) italic-ϕ 𝑥 {\displaystyle{\displaystyle\phi(x)}}

\phi(x)
Property / Symbols used: Q10916 / qualifier
 
xml-id: C1.S17.XMD1.m1cdec

Latest revision as of 16:43, 1 January 2020

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DLMF:1.17.E7
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    Statements

    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+).}}
    0 references
    DLMF:1.17.E7
    0 references
    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2abdec
    0 references
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    C1.S4.SS4.m1acdec
    0 references
    e {\displaystyle{\displaystyle\mathrm{e}}}
    C4.S2.E11.m2abdec
    0 references
    {\displaystyle{\displaystyle\int}}
    C1.S4.SS4.m3acdec
    0 references
    n 𝑛 {\displaystyle{\displaystyle n}}
    C1.S1.XMD7.m1ddec
    0 references
    ϕ ( x ) italic-ϕ 𝑥 {\displaystyle{\displaystyle\phi(x)}}
    C1.S17.XMD1.m1cdec
    0 references