DLMF:9.13.E10 (Q2972): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
(9 intermediate revisions by the same user not shown)
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2acdec
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
Defining formula:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2adec
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.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10901 / rank
 
Normal rank
Property / Symbols used: Q10901 / qualifier
 
Defining formula:

ℑ ⁡ absent {\displaystyle{\displaystyle\Im}}

\imagpart@@
Property / Symbols used: Q10901 / qualifier
 
xml-id: C1.S9.E2.m2adec
Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
Defining formula:

ph phase {\displaystyle{\displaystyle\operatorname{ph}}}

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.m1aadec
Property / Symbols used
 
Property / Symbols used: Q11381 / rank
 
Normal rank
Property / Symbols used: Q11381 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11381 / qualifier
 
xml-id: C9.S1.XMD3.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11386 / rank
 
Normal rank
Property / Symbols used: Q11386 / qualifier
 
Defining formula:

δ 𝛿 {\displaystyle{\displaystyle\delta}}

\delta
Property / Symbols used: Q11386 / qualifier
 
xml-id: C9.S1.XMD4.m1adec
Property / Symbols used
 
Property / Symbols used: parameter (locally) / rank
 
Normal rank
Property / Symbols used: parameter (locally) / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: parameter (locally) / qualifier
 
xml-id: C9.S13.XMD1.m1hdec
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.m1hdec
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.m1fdec

Latest revision as of 15:00, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:9.13.E10
No description defined

    Statements

    Defining formula
    A n ⁡ ( - z ) = { 2 ⁢ p / π ⁢ cos ⁡ ( 1 2 ⁢ p ⁢ π ) ⁢ z - n / 4 ⁢ ( cos ⁡ ( ζ - 1 4 ⁢ π ) + e | ℑ ⁡ ζ | ⁢ O ⁡ ( ζ - 1 ) ) , | ph ⁡ z | ≤ 2 ⁢ p ⁢ π - δ ,  n  odd , p / π ⁢ z - n / 4 ⁢ e ζ ⁢ ( 1 + O ⁡ ( ζ - 1 ) ) , | ph ⁡ z | ≤ p ⁢ π - δ ,  n  even , ODE-generalized-Airy-A 𝑛 𝑧 cases 2 𝑝 𝜋 1 2 𝑝 𝜋 superscript 𝑧 𝑛 4 𝜁 1 4 𝜋 superscript 𝑒 𝜁 Big-O superscript 𝜁 1 | ph ⁡ z | ≤ 2 ⁢ p ⁢ π - δ ,  n  odd 𝑝 𝜋 superscript 𝑧 𝑛 4 superscript 𝑒 𝜁 1 Big-O superscript 𝜁 1 | ph ⁡ z | ≤ p ⁢ π - δ ,  n  even {\displaystyle{\displaystyle A_{n}\left(-z\right)=\begin{cases}2\sqrt{p/\pi}% \cos\left(\tfrac{1}{2}p\pi\right)z^{-n/4}\left(\cos\left(\zeta-\tfrac{1}{4}\pi% \right)+e^{|\Im\zeta|}O\left(\zeta^{-1}\right)\right),&\text{$|\operatorname{% ph}z|\leq 2p\pi-\delta$, $n$ odd},\\ \sqrt{p/\pi}z^{-n/4}e^{\zeta}\left(1+O\left(\zeta^{-1}\right)\right),&\text{$|% \operatorname{ph}z|\leq p\pi-\delta$, $n$ even},\end{cases}}}
    0 references
    DLMF id
    DLMF:9.13.E10
    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.m2aadec
    0 references
    Q11416
    Defining formula
    A n ⁡ ( z ) ODE-generalized-Airy-A 𝑛 𝑧 {\displaystyle{\displaystyle A_{\NVar{n}}\left(\NVar{z}\right)}}
    xml-id
    C9.S13.SS1.p2.m3afdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    Q10901
    Defining formula
    ℑ ⁡ absent {\displaystyle{\displaystyle\Im}}
    xml-id
    C1.S9.E2.m2adec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aadec
    0 references
    Q11381
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C9.S1.XMD3.m1hdec
    0 references
    Q11386
    Defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C9.S1.XMD4.m1adec
    0 references
    parameter (locally)
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C9.S13.XMD1.m1hdec
    0 references
    Q11414
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C9.S13.XMD3.m1hdec
    0 references
    Q11415
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C9.S13.XMD4.m1fdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q2972&oldid=53182"