DLMF:8.11.E5 (Q2576): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2abdec

Revision as of 12:53, 2 January 2020

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DLMF:8.11.E5
No description defined

    Statements

    test
    P ⁡ ( a , z ) ∼ z a ⁢ e - z Γ ⁡ ( 1 + a ) ∼ ( 2 ⁢ π ⁢ a ) - 1 2 ⁢ e a - z ⁢ ( z / a ) a , asymptotically-equals incomplete-gamma-P 𝑎 𝑧 superscript 𝑧 𝑎 superscript 𝑒 𝑧 Euler-Gamma 1 𝑎 asymptotically-equals superscript 2 𝜋 𝑎 1 2 superscript 𝑒 𝑎 𝑧 superscript 𝑧 𝑎 𝑎 {\displaystyle{\displaystyle P\left(a,z\right)\sim\frac{z^{a}e^{-z}}{\Gamma% \left(1+a\right)}\sim(2\pi a)^{-\frac{1}{2}}e^{a-z}(z/a)^{a},}}
    0 references
    DLMF id
    DLMF:8.11.E5
    0 references
    constraint
    | ph ⁡ a | ≤ π - δ phase 𝑎 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}a|\leq\pi-\delta}}
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    a → ∞ → 𝑎 {\displaystyle{\displaystyle a\to\infty}}
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    | ph ⁡ a | ≤ π - δ ph 𝑎 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}a|\leq\pi-\delta}}
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    asymptotic equality
    test
    ∼ asymptotically-equals {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.E1.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
     
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