DLMF:12.14.E26 (Q4266): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
(8 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.m2agdec
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.m2acdec
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.m2aedec
Property / Symbols used
 
Property / Symbols used: Q11543 / rank
 
Normal rank
Property / Symbols used: Q11543 / qualifier
 
Defining formula:

W ⁡ ( a , x ) parabolic-W 𝑎 𝑥 {\displaystyle{\displaystyle W\left(\NVar{a},\NVar{x}\right)}}

\paraW@{\NVar{a}}{\NVar{x}}
Property / Symbols used: Q11543 / qualifier
 
xml-id: C12.S14.SS1.p1.m1andec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2acdec
Property / Symbols used
 
Property / Symbols used: Q11526 / rank
 
Normal rank
Property / Symbols used: Q11526 / qualifier
 
Defining formula:

s 𝑠 {\displaystyle{\displaystyle s}}

s
Property / Symbols used: Q11526 / qualifier
 
xml-id: C12.S1.XMD5.m1adec
Property / Symbols used
 
Property / Symbols used: coefficients (locally) / rank
 
Normal rank
Property / Symbols used: coefficients (locally) / qualifier
 
Defining formula:

𝒜 s ⁢ ( t ) subscript 𝒜 𝑠 𝑡 {\displaystyle{\displaystyle\mathcal{A}_{s}(t)}}

\mathcal{A}_{s}(t)
Property / Symbols used: coefficients (locally) / qualifier
 
xml-id: C12.S10.XMD2.m1adec
Property / Symbols used
 
Property / Symbols used: DLMF:12.14.E27 / rank
 
Normal rank
Property / Symbols used: DLMF:12.14.E27 / qualifier
 
Defining formula:

σ 𝜎 {\displaystyle{\displaystyle\sigma}}

\sigma
Property / Symbols used: DLMF:12.14.E27 / qualifier
 
xml-id: C12.S14.XMD12.m1adec
Property / Symbols used
 
Property / Symbols used: DLMF:12.14.E28 / rank
 
Normal rank
Property / Symbols used: DLMF:12.14.E28 / qualifier
 
Defining formula:

l ⁢ ( μ ) 𝑙 𝜇 {\displaystyle{\displaystyle l(\mu)}}

l(\mu)
Property / Symbols used: DLMF:12.14.E28 / qualifier
 
xml-id: C12.S14.XMD13.m1adec

Latest revision as of 15:31, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:12.14.E26
No description defined

    Statements

    Defining formula
    W ⁡ ( 1 2 ⁢ μ 2 , - μ ⁢ t ⁢ 2 ) ∼ 2 1 2 ⁢ e 1 4 ⁢ π ⁢ μ 2 ⁢ l ⁢ ( μ ) ( t 2 - 1 ) 1 4 ⁢ ( sin ⁡ σ ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ 𝒜 2 ⁢ s ⁢ ( t ) μ 4 ⁢ s + cos ⁡ σ ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ 𝒜 2 ⁢ s + 1 ⁢ ( t ) μ 4 ⁢ s + 2 ) , asymptotic-expansion parabolic-W 1 2 superscript 𝜇 2 𝜇 𝑡 2 superscript 2 1 2 superscript 𝑒 1 4 𝜋 superscript 𝜇 2 𝑙 𝜇 superscript superscript 𝑡 2 1 1 4 𝜎 superscript subscript 𝑠 0 superscript 1 𝑠 subscript 𝒜 2 𝑠 𝑡 superscript 𝜇 4 𝑠 𝜎 superscript subscript 𝑠 0 superscript 1 𝑠 subscript 𝒜 2 𝑠 1 𝑡 superscript 𝜇 4 𝑠 2 {\displaystyle{\displaystyle W\left(\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right)% \sim\frac{2^{\frac{1}{2}}e^{\frac{1}{4}\pi\mu^{2}}l(\mu)}{(t^{2}-1)^{\frac{1}{% 4}}}\left(\sin\sigma\sum_{s=0}^{\infty}(-1)^{s}\frac{{\cal{A}}_{2s}(t)}{\mu^{4% s}}+\cos\sigma\sum_{s=0}^{\infty}(-1)^{s}\frac{{\cal{A}}_{2s+1}(t)}{\mu^{4s+2}% }\right),}}
    0 references
    DLMF id
    DLMF:12.14.E26
    0 references
    Symbols used
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11addec
    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
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    Q11543
    Defining formula
    W ⁡ ( a , x ) parabolic-W 𝑎 𝑥 {\displaystyle{\displaystyle W\left(\NVar{a},\NVar{x}\right)}}
    xml-id
    C12.S14.SS1.p1.m1andec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
    0 references
    Q11526
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C12.S1.XMD5.m1adec
    0 references
    coefficients (locally)
    Defining formula
    𝒜 s ⁢ ( t ) subscript 𝒜 𝑠 𝑡 {\displaystyle{\displaystyle\mathcal{A}_{s}(t)}}
    xml-id
    C12.S10.XMD2.m1adec
    0 references
    DLMF:12.14.E27
    Defining formula
    σ 𝜎 {\displaystyle{\displaystyle\sigma}}
    xml-id
    C12.S14.XMD12.m1adec
    0 references
    DLMF:12.14.E28
    Defining formula
    l ⁢ ( μ ) 𝑙 𝜇 {\displaystyle{\displaystyle l(\mu)}}
    xml-id
    C12.S14.XMD13.m1adec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q4266&oldid=58443"