DLMF:2.3.E23 (Q742): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

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

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1amdec
Property / Symbols used
 
Property / Symbols used: exponential function / rank
 
Normal rank
Property / Symbols used: exponential function / qualifier
 
test:

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

\exp@@{\NVar{z}}
Property / Symbols used: exponential function / qualifier
 
xml-id: C4.S2.E19.m2acdec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

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

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2akdec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
test:

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

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2aedec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3amdec
Property / Symbols used
 
Property / Symbols used: Q10958 / rank
 
Normal rank
Property / Symbols used: Q10958 / qualifier
 
test:

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

p(t)
Property / Symbols used: Q10958 / qualifier
 
xml-id: C2.S3.XMD17.m1hdec
Property / Symbols used
 
Property / Symbols used: Q10961 / rank
 
Normal rank
Property / Symbols used: Q10961 / qualifier
 
test:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q10961 / qualifier
 
xml-id: C2.S3.XMD23.m1adec
Property / Symbols used
 
Property / Symbols used: Q10962 / rank
 
Normal rank
Property / Symbols used: Q10962 / qualifier
 
test:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q10962 / qualifier
 
xml-id: C2.S3.XMD24.m1adec
Property / Symbols used
 
Property / Symbols used: function (locally) / rank
 
Normal rank
Property / Symbols used: function (locally) / qualifier
 
test:

P s ⁢ ( t ) subscript 𝑃 𝑠 𝑡 {\displaystyle{\displaystyle P_{s}(t)}}

P_{s}(t)
Property / Symbols used: function (locally) / qualifier
 
xml-id: C2.S3.XMD26.m1bdec
Property / Symbols used
 
Property / Symbols used: function (locally) / rank
 
Normal rank
Property / Symbols used: function (locally) / qualifier
 
test:

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

q(t)
Property / Symbols used: function (locally) / qualifier
 
xml-id: C2.S3.XMD6.m1hdec
Property / Symbols used
 
Property / Symbols used: Q10950 / rank
 
Normal rank
Property / Symbols used: Q10950 / qualifier
 
test:

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

\lambda
Property / Symbols used: Q10950 / qualifier
 
xml-id: C2.S3.XMD8.m1jdec
Property / Symbols used
 
Property / Symbols used: Q10951 / rank
 
Normal rank
Property / Symbols used: Q10951 / qualifier
 
test:

μ 𝜇 {\displaystyle{\displaystyle\mu}}

\mu
Property / Symbols used: Q10951 / qualifier
 
xml-id: C2.S3.XMD9.m1jdec

Latest revision as of 15:54, 1 January 2020

No description defined
Language Label Description Also known as
English
DLMF:2.3.E23
No description defined

    Statements

    test
    ∫ a b e i ⁢ x ⁢ p ⁢ ( t ) ⁢ q ⁢ ( t ) ⁢ d t ∼ e i ⁢ x ⁢ p ⁢ ( a ) ⁢ ∑ s = 0 ∞ exp ⁡ ( ( s + λ ) ⁢ π ⁢ i 2 ⁢ μ ) ⁢ Γ ⁡ ( s + λ μ ) ⁢ b s x ( s + λ ) / μ - e i ⁢ x ⁢ p ⁢ ( b ) ⁢ ∑ s = 0 ∞ P s ⁢ ( b ) ⁢ ( i x ) s + 1 , asymptotic-expansion superscript subscript 𝑎 𝑏 superscript 𝑒 𝑖 𝑥 𝑝 𝑡 𝑞 𝑡 𝑡 superscript 𝑒 𝑖 𝑥 𝑝 𝑎 superscript subscript 𝑠 0 𝑠 𝜆 𝜋 𝑖 2 𝜇 Euler-Gamma 𝑠 𝜆 𝜇 subscript 𝑏 𝑠 superscript 𝑥 𝑠 𝜆 𝜇 superscript 𝑒 𝑖 𝑥 𝑝 𝑏 superscript subscript 𝑠 0 subscript 𝑃 𝑠 𝑏 superscript 𝑖 𝑥 𝑠 1 {\displaystyle{\displaystyle\int_{a}^{b}e^{ixp(t)}q(t)\mathrm{d}t\sim e^{ixp(a% )}\sum_{s=0}^{\infty}\exp\left(\frac{(s+\lambda)\pi i}{2\mu}\right)\Gamma\left% (\frac{s+\lambda}{\mu}\right)\frac{b_{s}}{x^{(s+\lambda)/\mu}}-e^{ixp(b)}\sum_% {s=0}^{\infty}P_{s}(b)\left(\frac{i}{x}\right)^{s+1},}}
    0 references
    DLMF id
    DLMF:2.3.E23
    0 references
    constraint
    x → + ∞ → 𝑥 {\displaystyle{\displaystyle x\to+\infty}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
    Q10931
    test
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11akdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1amdec
    0 references
    exponential function
    test
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2acdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2akdec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3amdec
    0 references
    Q10958
    test
    p ⁢ ( t ) 𝑝 𝑡 {\displaystyle{\displaystyle p(t)}}
    xml-id
    C2.S3.XMD17.m1hdec
    0 references
    Q10961
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C2.S3.XMD23.m1adec
    0 references
    Q10962
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C2.S3.XMD24.m1adec
    0 references
    function (locally)
    test
    P s ⁢ ( t ) subscript 𝑃 𝑠 𝑡 {\displaystyle{\displaystyle P_{s}(t)}}
    xml-id
    C2.S3.XMD26.m1bdec
    0 references
    function (locally)
    test
    q ⁢ ( t ) 𝑞 𝑡 {\displaystyle{\displaystyle q(t)}}
    xml-id
    C2.S3.XMD6.m1hdec
    0 references
    Q10950
    test
    λ 𝜆 {\displaystyle{\displaystyle\lambda}}
    xml-id
    C2.S3.XMD8.m1jdec
    0 references
    Q10951
    test
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C2.S3.XMD9.m1jdec
    0 references
     
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