DLMF:8.6.E12 (Q2526)

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DLMF:8.6.E12
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    Statements

    test
    Γ ⁡ ( a , z ) = - z a - 1 ⁢ e - z Γ ⁡ ( 1 - a ) ⁢ 1 2 ⁢ π ⁢ i ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ Γ ⁡ ( s + 1 - a ) ⁢ π ⁢ z - s sin ⁡ ( π ⁢ s ) ⁢ d s , incomplete-Gamma 𝑎 𝑧 superscript 𝑧 𝑎 1 superscript 𝑒 𝑧 Euler-Gamma 1 𝑎 1 2 𝜋 𝑖 superscript subscript 𝑐 𝑖 𝑐 𝑖 Euler-Gamma 𝑠 1 𝑎 𝜋 superscript 𝑧 𝑠 𝜋 𝑠 𝑠 {\displaystyle{\displaystyle\Gamma\left(a,z\right)=-\frac{z^{a-1}e^{-z}}{% \Gamma\left(1-a\right)}\*\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma% \left(s+1-a\right)\frac{\pi z^{-s}}{\sin\left(\pi s\right)}\mathrm{d}s,}}
    0 references
    DLMF id
    DLMF:8.6.E12
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    constraint
    | ph ⁡ z | < 3 2 ⁢ π phase 𝑧 3 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{3}{2}\pi}}
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    | ph ⁡ z | < 3 2 ⁢ π ph 𝑧 3 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{3}{2}\pi}}
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    a ≠ 1 , 2 , 3 , … 𝑎 1 2 3 … {\displaystyle{\displaystyle a\neq 1,2,3,\dots}}
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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.m2aedec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1akdec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aidec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
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    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.m2afdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3akdec
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    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1acdec
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    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
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    Q11323
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C8.S1.XMD2.m1kdec
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    Q11324
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1kdec
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    Q11331
    test
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C8.S6.XMD1.m1bdec
    0 references
     
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