DLMF:26.7.E7 (Q7824): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q12172 / rank
 
Normal rank
Property / Symbols used: Q12172 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12172 / qualifier
 
xml-id: C26.S1.XMD6.m1fdec
Property / Symbols used
 
Property / Symbols used: Q12185 / rank
 
Normal rank
Property / Symbols used: Q12185 / qualifier
 
Defining formula:

N 𝑁 {\displaystyle{\displaystyle N}}

N
Property / Symbols used: Q12185 / qualifier
 
xml-id: C26.S7.XMD1.m1dec

Latest revision as of 01:13, 2 January 2020

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DLMF:26.7.E7
No description defined

    Statements

    Defining formula
    B ⁡ ( n ) = N n ⁢ e N - n - 1 ( 1 + ln ⁡ N ) 1 / 2 ⁢ ( 1 + O ⁡ ( ( ln ⁡ n ) 1 / 2 n 1 / 2 ) ) , Bell-number 𝑛 superscript 𝑁 𝑛 𝑁 𝑛 1 superscript 1 𝑁 1 2 1 Big-O superscript 𝑛 1 2 superscript 𝑛 1 2 {\displaystyle{\displaystyle B\left(n\right)=\frac{N^{n}{\mathrm{e}^{N-n-1}}}{% (1+\ln N)^{1/2}}\left(1+O\left(\frac{(\ln n)^{1/2}}{n^{1/2}}\right)\right),}}
    0 references
    DLMF id
    DLMF:26.7.E7
    0 references
    constraint
    n → ∞ → 𝑛 {\displaystyle{\displaystyle n\to\infty}}
    0 references
    Symbols used
    Q12184
    Defining formula
    B ⁡ ( n ) Bell-number 𝑛 {\displaystyle{\displaystyle B\left(\NVar{n}\right)}}
    xml-id
    C26.S7.SS1.m1agdec
    0 references
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    Q12172
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C26.S1.XMD6.m1fdec
    0 references
    Q12185
    Defining formula
    N 𝑁 {\displaystyle{\displaystyle N}}
    xml-id
    C26.S7.XMD1.m1dec
    0 references
     
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