DLMF:8.6.E9 (Q2523): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3ahdec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1afdec
Property / Symbols used
 
Property / Symbols used: Q11323 / rank
 
Normal rank
Property / Symbols used: Q11323 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11323 / qualifier
 
xml-id: C8.S1.XMD2.m1hdec

Latest revision as of 12:44, 2 January 2020

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English
DLMF:8.6.E9
No description defined

    Statements

    test
    Γ ⁡ ( - a , z ⁢ e ± π ⁢ i ) = e z ⁢ e ∓ π ⁢ i ⁢ a Γ ⁡ ( 1 + a ) ⁢ ∫ 0 ∞ t a ⁢ e - z ⁢ t t - 1 ⁢ d t , incomplete-Gamma 𝑎 𝑧 superscript 𝑒 plus-or-minus 𝜋 𝑖 superscript 𝑒 𝑧 superscript 𝑒 minus-or-plus 𝜋 imaginary-unit 𝑎 Euler-Gamma 1 𝑎 superscript subscript 0 superscript 𝑡 𝑎 superscript 𝑒 𝑧 𝑡 𝑡 1 𝑡 {\displaystyle{\displaystyle\Gamma\left(-a,ze^{\pm\pi i}\right)=\frac{e^{z}e^{% \mp\pi\mathrm{i}a}}{\Gamma\left(1+a\right)}\int_{0}^{\infty}\frac{t^{a}e^{-zt}% }{t-1}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:8.6.E9
    0 references
    constraint
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
    0 references
    ℜ ⁡ a > - 1 𝑎 1 {\displaystyle{\displaystyle\Re a>-1}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ahdec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    incomplete gamma function
    test
    Γ ⁡ ( a , z ) incomplete-Gamma 𝑎 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E2.m2addec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ahdec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1afdec
    0 references
    Q11323
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C8.S1.XMD2.m1hdec
    0 references
     
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