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DLMF:8.6.E8 - MaRDI portal

DLMF:8.6.E8 (Q3220)

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DLMF:8.6.E8
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    Statements

    DLMF defining formula
    γ ⁡ ( a , z ) = - i ⁢ z a 2 ⁢ sin ⁡ ( π ⁢ a ) ⁢ ∫ - 1 ( 0 + ) t a - 1 ⁢ e z ⁢ t ⁢ d t , incomplete-gamma 𝑎 𝑧 imaginary-unit superscript 𝑧 𝑎 2 𝜋 𝑎 superscript subscript 1 limit-from 0 superscript 𝑡 𝑎 1 superscript 𝑒 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\gamma\left(a,z\right)=\frac{-\mathrm{i}z^{a}}{2% \sin\left(\pi a\right)}\int_{-1}^{(0+)}t^{a-1}e^{zt}\mathrm{d}t,}}
    0 references
    constraint
    a ∉ ℤ 𝑎 {\displaystyle{\displaystyle a\notin\mathbb{Z}}}
    0 references
    z ≠ 0 𝑧 0 {\displaystyle{\displaystyle z\neq 0}}
    0 references
    a ∉ ℤ 𝑎 ℤ {\displaystyle{\displaystyle a\notin\mathbb{Z}}}
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1agdec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    incomplete gamma function
    DLMF defining formula
    γ ⁡ ( a , z ) incomplete-gamma 𝑎 𝑧 {\displaystyle{\displaystyle\gamma\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E1.m2acdec
    0 references
    Q10792
    DLMF defining formula
    ℤ {\displaystyle{\displaystyle\mathbb{Z}}}
    xml-id
    introduction.Sx4.p2.t1.r20.m2aadec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3agdec
    0 references
    Q11216
    DLMF defining formula
    ∉ {\displaystyle{\displaystyle\notin}}
    xml-id
    introduction.Sx4.p1.t1.r11.m2aadec
    0 references
    sine function
    DLMF defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C8.S1.XMD2.m1gdec
    0 references
    parameter
    DLMF defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1gdec
    0 references
     
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