DLMF:8.11.E12 (Q2585): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: incomplete gamma function / rank
 
Normal rank
Property / Symbols used: incomplete gamma function / qualifier
 
Defining formula:

Γ ⁡ ( a , z ) incomplete-Gamma 𝑎 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{a},\NVar{z}\right)}}

\incGamma@{\NVar{a}}{\NVar{z}}
Property / Symbols used: incomplete gamma function / qualifier
 
xml-id: C8.S2.E2.m2abdec

Revision as of 13:55, 2 January 2020

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DLMF:8.11.E12
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    Statements

    Defining formula
    Γ ⁡ ( z , z ) ∼ z z - 1 ⁢ e - z ⁢ ( π 2 ⁢ z 1 2 - 1 3 + 2 ⁢ π 24 ⁢ z 1 2 - 4 135 ⁢ z + 2 ⁢ π 576 ⁢ z 3 2 + 8 2835 ⁢ z 2 + … ) , asymptotic-expansion incomplete-Gamma 𝑧 𝑧 superscript 𝑧 𝑧 1 superscript 𝑒 𝑧 𝜋 2 superscript 𝑧 1 2 1 3 2 𝜋 24 superscript 𝑧 1 2 4 135 𝑧 2 𝜋 576 superscript 𝑧 3 2 8 2835 superscript 𝑧 2 … {\displaystyle{\displaystyle\Gamma\left(z,z\right)\sim z^{z-1}e^{-z}\left(% \sqrt{\frac{\pi}{2}}z^{\frac{1}{2}}-\frac{1}{3}+\frac{\sqrt{2\pi}}{24z^{\frac{% 1}{2}}}-\frac{4}{135z}+\frac{\sqrt{2\pi}}{576z^{\frac{3}{2}}}+\frac{8}{2835z^{% 2}}+\dots\right),}}
    0 references
    DLMF id
    DLMF:8.11.E12
    0 references
    constraint
    | ph ⁡ z | ≤ 2 ⁢ π - δ phase 𝑧 2 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq 2\pi-\delta}}
    0 references
    Symbols used
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
    0 references
    incomplete gamma function
    Defining formula
    Γ ⁡ ( a , z ) incomplete-Gamma 𝑎 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E2.m2abdec
    0 references
     
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