DLMF:9.13.E12 (Q2974): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
(One intermediate revision by the same user not shown)
Property / Symbols used
 
Property / Symbols used: Q11414 / rank
 
Normal rank
Property / Symbols used: Q11414 / qualifier
 
Defining formula:

p 𝑝 {\displaystyle{\displaystyle p}}

p
Property / Symbols used: Q11414 / qualifier
 
xml-id: C9.S13.XMD3.m1jdec
Property / Symbols used
 
Property / Symbols used: Q11415 / rank
 
Normal rank
Property / Symbols used: Q11415 / qualifier
 
Defining formula:

ζ 𝜁 {\displaystyle{\displaystyle\zeta}}

\zeta
Property / Symbols used: Q11415 / qualifier
 
xml-id: C9.S13.XMD4.m1hdec

Latest revision as of 15:01, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:9.13.E12
No description defined

    Statements

    Defining formula
    B n ⁡ ( - z ) = { - ( 2 / π ) ⁢ sin ⁡ ( 1 2 ⁢ p ⁢ π ) ⁢ z - n / 4 ⁢ ( sin ⁡ ( ζ - 1 4 ⁢ π ) + e | ℑ ⁡ ζ | ⁢ O ⁡ ( ζ - 1 ) ) , | ph ⁡ z | ≤ 2 ⁢ p ⁢ π - δ , n ⁢  odd , ( 1 / π ) ⁢ sin ⁡ ( p ⁢ π ) ⁢ z - n / 4 ⁢ e - ζ ⁢ ( 1 + O ⁡ ( ζ - 1 ) ) , | ph ⁡ z | ≤ 3 ⁢ p ⁢ π - δ , n ⁢  even . ODE-generalized-Airy-B 𝑛 𝑧 cases 2 𝜋 1 2 𝑝 𝜋 superscript 𝑧 𝑛 4 𝜁 1 4 𝜋 superscript 𝑒 𝜁 Big-O superscript 𝜁 1 phase 𝑧 2 𝑝 𝜋 𝛿 𝑛  odd 1 𝜋 𝑝 𝜋 superscript 𝑧 𝑛 4 superscript 𝑒 𝜁 1 Big-O superscript 𝜁 1 phase 𝑧 3 𝑝 𝜋 𝛿 𝑛  even {\displaystyle{\displaystyle B_{n}\left(-z\right)=\begin{cases}-(\ifrac{2}{% \sqrt{\pi}})\sin\left(\tfrac{1}{2}p\pi\right)z^{-n/4}\left(\sin\left(\zeta-% \tfrac{1}{4}\pi\right)+e^{\left|\Im\zeta\right|}O\left(\zeta^{-1}\right)\right% ),&\left|\operatorname{ph}z\right|\leq 2p\pi-\delta,n\text{ odd},\\ (\ifrac{1}{\sqrt{\pi}})\sin\left(p\pi\right)z^{-n/4}e^{-\zeta}\left(1+O\left(% \zeta^{-1}\right)\right),&\left|\operatorname{ph}z\right|\leq 3p\pi-\delta,n% \text{ even}.\end{cases}}}
    0 references
    DLMF id
    DLMF:9.13.E12
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2acdec
    0 references
    Q11417
    Defining formula
    B n ⁡ ( z ) ODE-generalized-Airy-B 𝑛 𝑧 {\displaystyle{\displaystyle B_{\NVar{n}}\left(\NVar{z}\right)}}
    xml-id
    C9.S13.SS1.p2.m4afdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    Q10901
    Defining formula
    ℑ ⁡ absent {\displaystyle{\displaystyle\Im}}
    xml-id
    C1.S9.E2.m2aadec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1acdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
    0 references
    Q11381
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1jdec
    0 references
    Q11386
    Defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C9.S1.XMD4.m1cdec
    0 references
    parameter (locally)
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C9.S13.XMD1.m1jdec
    0 references
    Q11414
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C9.S13.XMD3.m1jdec
    0 references
    Q11415
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C9.S13.XMD4.m1hdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q2974&oldid=53321"