DLMF:2.4.E15 (Q769): Difference between revisions

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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.m2akdec
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.m3ahdec
Property / Symbols used
 
Property / Symbols used: Q10972 / rank
 
Normal rank
Property / Symbols used: Q10972 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q10972 / qualifier
 
xml-id: C2.S4.XMD11.m1fdec
Property / Symbols used
 
Property / Symbols used: Q10980 / rank
 
Normal rank
Property / Symbols used: Q10980 / qualifier
 
Defining formula:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q10980 / qualifier
 
xml-id: C2.S4.XMD12.m1cdec
Property / Symbols used
 
Property / Symbols used: Q10981 / rank
 
Normal rank
Property / Symbols used: Q10981 / qualifier
 
Defining formula:

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

p(t)
Property / Symbols used: Q10981 / qualifier
 
xml-id: C2.S4.XMD15.m1ddec
Property / Symbols used
 
Property / Symbols used: Q10973 / rank
 
Normal rank
Property / Symbols used: Q10973 / qualifier
 
Defining formula:

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

q(t)
Property / Symbols used: Q10973 / qualifier
 
xml-id: C2.S4.XMD16.m1jdec

Latest revision as of 16:59, 1 January 2020

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English
DLMF:2.4.E15
No description defined

    Statements

    Defining formula
    ∫ a b e - z ⁢ p ⁢ ( t ) ⁢ q ⁢ ( t ) ⁢ d t ∼ 2 ⁢ e - z ⁢ p ⁢ ( t 0 ) ⁢ ∑ s = 0 ∞ Γ ⁡ ( s + 1 2 ) ⁢ b 2 ⁢ s z s + ( 1 / 2 ) , asymptotic-expansion superscript subscript 𝑎 𝑏 superscript 𝑒 𝑧 𝑝 𝑡 𝑞 𝑡 𝑡 2 superscript 𝑒 𝑧 𝑝 subscript 𝑡 0 superscript subscript 𝑠 0 Euler-Gamma 𝑠 1 2 subscript 𝑏 2 𝑠 superscript 𝑧 𝑠 1 2 {\displaystyle{\displaystyle\int_{a}^{b}e^{-zp(t)}q(t)\mathrm{d}t\sim 2e^{-zp(% t_{0})}\sum_{s=0}^{\infty}\Gamma\left(s+\tfrac{1}{2}\right)\frac{b_{2s}}{z^{s+% (1/2)}},}}
    0 references
    DLMF id
    DLMF:2.4.E15
    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.m11acdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2akdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ahdec
    0 references
    Q10972
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C2.S4.XMD11.m1fdec
    0 references
    Q10980
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C2.S4.XMD12.m1cdec
    0 references
    Q10981
    Defining formula
    p ⁢ ( t ) 𝑝 𝑡 {\displaystyle{\displaystyle p(t)}}
    xml-id
    C2.S4.XMD15.m1ddec
    0 references
    Q10973
    Defining formula
    q ⁢ ( t ) 𝑞 𝑡 {\displaystyle{\displaystyle q(t)}}
    xml-id
    C2.S4.XMD16.m1jdec
    0 references
     
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