DLMF:2.3.E4 (Q720): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10945 / rank
 
Normal rank
Property / Symbols used: Q10945 / qualifier
 
Defining formula:

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

q(t)
Property / Symbols used: Q10945 / qualifier
 
xml-id: C2.S3.XMD1.m1cdec
Property / Symbols used
 
Property / Symbols used: Q10947 / rank
 
Normal rank
Property / Symbols used: Q10947 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q10947 / qualifier
 
xml-id: C2.S3.XMD4.m1dec
Property / Symbols used
 
Property / Symbols used: Q10948 / rank
 
Normal rank
Property / Symbols used: Q10948 / qualifier
 
Defining formula:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q10948 / qualifier
 
xml-id: C2.S3.XMD5.m1dec

Latest revision as of 16:50, 1 January 2020

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DLMF:2.3.E4
No description defined

    Statements

    Defining formula
    ∫ a b e i ⁢ x ⁢ t ⁢ q ⁢ ( t ) ⁢ d t ∼ e i ⁢ a ⁢ x ⁢ ∑ s = 0 ∞ q ( s ) ⁢ ( a ) ⁢ ( i x ) s + 1 - e i ⁢ b ⁢ x ⁢ ∑ s = 0 ∞ q ( s ) ⁢ ( b ) ⁢ ( i x ) s + 1 , asymptotic-expansion superscript subscript 𝑎 𝑏 superscript 𝑒 𝑖 𝑥 𝑡 𝑞 𝑡 𝑡 superscript 𝑒 𝑖 𝑎 𝑥 superscript subscript 𝑠 0 superscript 𝑞 𝑠 𝑎 superscript 𝑖 𝑥 𝑠 1 superscript 𝑒 𝑖 𝑏 𝑥 superscript subscript 𝑠 0 superscript 𝑞 𝑠 𝑏 superscript 𝑖 𝑥 𝑠 1 {\displaystyle{\displaystyle\int_{a}^{b}e^{ixt}q(t)\mathrm{d}t\sim e^{iax}\sum% _{s=0}^{\infty}q^{(s)}(a)\left(\frac{i}{x}\right)^{s+1}-e^{ibx}\sum_{s=0}^{% \infty}q^{(s)}(b)\left(\frac{i}{x}\right)^{s+1},}}
    0 references
    DLMF id
    DLMF:2.3.E4
    0 references
    constraint
    x → + ∞ → 𝑥 {\displaystyle{\displaystyle x\to+\infty}}
    0 references
    Symbols used
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    Q10945
    Defining formula
    q ⁢ ( t ) 𝑞 𝑡 {\displaystyle{\displaystyle q(t)}}
    xml-id
    C2.S3.XMD1.m1cdec
    0 references
    Q10947
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C2.S3.XMD4.m1dec
    0 references
    Q10948
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C2.S3.XMD5.m1dec
    0 references
     
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