DLMF:8.12.E16 (Q2618): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2addec

Revision as of 13:59, 2 January 2020

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English
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}}
    0 references
    Symbols used
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
     
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