DLMF:8.21.E3 (Q2724): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11324 / rank
 
Normal rank
Property / Symbols used: Q11324 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11324 / qualifier
 
xml-id: C8.S1.XMD3.m1bdec

Latest revision as of 14:15, 2 January 2020

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DLMF:8.21.E3
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    Statements

    Defining formula
    ∫ 0 ∞ t a - 1 ⁢ e ± i ⁢ t ⁢ d t = e ± 1 2 ⁢ π ⁢ i ⁢ a ⁢ Γ ⁡ ( a ) , superscript subscript 0 superscript 𝑡 𝑎 1 superscript 𝑒 plus-or-minus imaginary-unit 𝑡 𝑡 superscript 𝑒 plus-or-minus 1 2 𝜋 imaginary-unit 𝑎 Euler-Gamma 𝑎 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{a-1}e^{\pm\mathrm{i}t}\mathrm{% d}t=e^{\pm\frac{1}{2}\pi\mathrm{i}a}\Gamma\left(a\right),}}
    0 references
    DLMF id
    DLMF:8.21.E3
    0 references
    constraint
    0 < ℜ ⁡ a < 1 0 𝑎 1 {\displaystyle{\displaystyle 0<\Re a<1}}
    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.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    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.m2abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    Q11324
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S1.XMD3.m1bdec
    0 references
     
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