DLMF:8.12.E5 (Q2599)

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DLMF:8.12.E5
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    Statements

    Defining formula
    e ± π ⁢ i ⁢ a 2 ⁢ i ⁢ sin ⁡ ( π ⁢ a ) ⁢ Q ⁡ ( - a , z ⁢ e ± π ⁢ i ) = ± 1 2 ⁢ erfc ⁡ ( ± i ⁢ η ⁢ a / 2 ) - i ⁢ T ⁢ ( a , η ) , superscript 𝑒 plus-or-minus 𝜋 𝑖 𝑎 2 𝑖 𝜋 𝑎 incomplete-gamma-Q 𝑎 𝑧 superscript 𝑒 plus-or-minus 𝜋 𝑖 plus-or-minus 1 2 complementary-error-function plus-or-minus 𝑖 𝜂 𝑎 2 𝑖 𝑇 𝑎 𝜂 {\displaystyle{\displaystyle\frac{e^{\pm\pi ia}}{2i\sin\left(\pi a\right)}Q% \left(-a,ze^{\pm\pi i}\right)=\pm\tfrac{1}{2}\operatorname{erfc}\left(\pm i% \eta\sqrt{a/2}\right)-iT(a,\eta),}}
    0 references
    DLMF id
    DLMF:8.12.E5
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    complementary error function
    Defining formula
    erfc ⁡ z complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\NVar{z}}}
    xml-id
    C7.S2.E2.m2abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    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.m2adec
    0 references
    Q11326
    Defining formula
    Q ⁡ ( a , z ) incomplete-gamma-Q 𝑎 𝑧 {\displaystyle{\displaystyle Q\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C8.S2.E4.m2aadec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q11323
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C8.S1.XMD2.m1cdec
    0 references
    Q11324
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1cdec
    0 references
    Q11348
    Defining formula
    η ⁢ ( λ ) 𝜂 𝜆 {\displaystyle{\displaystyle\eta(\lambda)}}
    xml-id
    C8.S12.XMD1.m1ddec
    0 references
    Q11350
    Defining formula
    T ⁢ ( a , η ) 𝑇 𝑎 𝜂 {\displaystyle{\displaystyle T(a,\eta)}}
    xml-id
    C8.S12.XMD4.m1dec
    0 references
     
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