DLMF:8.12.E16 (Q2618)

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DLMF:8.12.E16
No description defined

    Statements

    Defining formula
    e ± π ⁢ i ⁢ a 2 ⁢ i ⁢ sin ⁡ ( π ⁢ a ) ⁢ Q ⁡ ( - a , a ⁢ e ± π ⁢ i ) ∼ ± 1 2 - i 2 ⁢ π ⁢ a ⁢ ∑ k = 0 ∞ c k ⁢ ( 0 ) ⁢ ( - a ) - k , asymptotic-expansion superscript 𝑒 plus-or-minus 𝜋 𝑖 𝑎 2 𝑖 𝜋 𝑎 incomplete-gamma-Q 𝑎 𝑎 superscript 𝑒 plus-or-minus 𝜋 𝑖 plus-or-minus 1 2 𝑖 2 𝜋 𝑎 superscript subscript 𝑘 0 subscript 𝑐 𝑘 0 superscript 𝑎 𝑘 {\displaystyle{\displaystyle\frac{e^{\pm\pi ia}}{2i\sin\left(\pi a\right)}Q% \left(-a,ae^{\pm\pi i}\right)\sim\pm\frac{1}{2}-\frac{i}{\sqrt{2\pi a}}\sum_{k% =0}^{\infty}c_{k}(0)(-a)^{-k},}}
    0 references
    DLMF id
    DLMF:8.12.E16
    0 references
    constraint
    | ph ⁡ a | ≤ π - δ phase 𝑎 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}a|\leq\pi-\delta}}
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    Symbols used
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11acdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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    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.m2acdec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aadec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
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    Q11324
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1hdec
    0 references
    Q11330
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C8.S1.XMD5.m1fdec
    0 references
    Q11343
    Defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C8.S1.XMD7.m1adec
    0 references
     
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