DLMF:2.3.E20 (Q739): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10931 / rank
 
Normal rank
Property / Symbols used: Q10931 / qualifier
 
Defining formula:

∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}

\asympexp
Property / Symbols used: Q10931 / qualifier
 
xml-id: C2.S1.SS3.p1.m11ajdec
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.m2adec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1akdec
Property / Symbols used
 
Property / Symbols used: exponential function / rank
 
Normal rank
Property / Symbols used: exponential function / qualifier
 
Defining formula:

exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}

\exp@@{\NVar{z}}
Property / Symbols used: exponential function / qualifier
 
xml-id: C4.S2.E19.m2abdec
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.m2aidec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2acdec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3akdec
Property / Symbols used
 
Property / Symbols used: function (locally) / rank
 
Normal rank
Property / Symbols used: function (locally) / qualifier
 
Defining formula:

q ⁢ ( t ) 𝑞 𝑡 {\displaystyle{\displaystyle q(t)}}

q(t)
Property / Symbols used: function (locally) / qualifier
 
xml-id: C2.S3.XMD6.m1fdec
Property / Symbols used
 
Property / Symbols used: Q10963 / rank
 
Normal rank
Property / Symbols used: Q10963 / qualifier
 
Defining formula:

a s subscript 𝑎 𝑠 {\displaystyle{\displaystyle a_{s}}}

a_{s}
Property / Symbols used: Q10963 / qualifier
 
xml-id: C2.S3.XMD7.m1dec
Property / Symbols used
 
Property / Symbols used: Q10950 / rank
 
Normal rank
Property / Symbols used: Q10950 / qualifier
 
Defining formula:

λ 𝜆 {\displaystyle{\displaystyle\lambda}}

\lambda
Property / Symbols used: Q10950 / qualifier
 
xml-id: C2.S3.XMD8.m1idec
Property / Symbols used
 
Property / Symbols used: Q10951 / rank
 
Normal rank
Property / Symbols used: Q10951 / qualifier
 
Defining formula:

μ 𝜇 {\displaystyle{\displaystyle\mu}}

\mu
Property / Symbols used: Q10951 / qualifier
 
xml-id: C2.S3.XMD9.m1idec

Latest revision as of 16:53, 1 January 2020

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DLMF:2.3.E20
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ e i ⁢ x ⁢ t ⁢ q ⁢ ( t ) ⁢ d t ∼ ∑ s = 0 ∞ exp ⁡ ( ( s + λ ) ⁢ π ⁢ i 2 ⁢ μ ) ⁢ Γ ⁡ ( s + λ μ ) ⁢ a s x ( s + λ ) / μ , asymptotic-expansion superscript subscript 0 superscript 𝑒 𝑖 𝑥 𝑡 𝑞 𝑡 𝑡 superscript subscript 𝑠 0 𝑠 𝜆 𝜋 𝑖 2 𝜇 Euler-Gamma 𝑠 𝜆 𝜇 subscript 𝑎 𝑠 superscript 𝑥 𝑠 𝜆 𝜇 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{ixt}q(t)\mathrm{d}t\sim\sum_{s% =0}^{\infty}\exp\left(\frac{(s+\lambda)\pi i}{2\mu}\right)\Gamma\left(\frac{s+% \lambda}{\mu}\right)\frac{a_{s}}{x^{(s+\lambda)/\mu}},}}
    0 references
    DLMF id
    DLMF:2.3.E20
    0 references
    constraint
    x → + ∞ → 𝑥 {\displaystyle{\displaystyle x\to+\infty}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11ajdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1akdec
    0 references
    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aidec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3akdec
    0 references
    function (locally)
    Defining formula
    q ⁢ ( t ) 𝑞 𝑡 {\displaystyle{\displaystyle q(t)}}
    xml-id
    C2.S3.XMD6.m1fdec
    0 references
    Q10963
    Defining formula
    a s subscript 𝑎 𝑠 {\displaystyle{\displaystyle a_{s}}}
    xml-id
    C2.S3.XMD7.m1dec
    0 references
    Q10950
    Defining formula
    λ 𝜆 {\displaystyle{\displaystyle\lambda}}
    xml-id
    C2.S3.XMD8.m1idec
    0 references
    Q10951
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C2.S3.XMD9.m1idec
    0 references
     
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