DLMF:30.15.E5 (Q8947)

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DLMF:30.15.E5
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    Statements

    Defining formula
    ∫ - τ τ e - i ⁢ t ⁢ ω ⁢ ϕ n ⁢ ( t ) ⁢ d t = ( - i ) n ⁢ 2 ⁢ π ⁢ τ ⁢ Λ n σ ⁢ ϕ n ⁢ ( τ σ ⁢ ω ) , superscript subscript 𝜏 𝜏 superscript 𝑒 imaginary-unit 𝑡 𝜔 subscript italic-ϕ 𝑛 𝑡 𝑡 superscript imaginary-unit 𝑛 2 𝜋 𝜏 subscript Λ 𝑛 𝜎 subscript italic-ϕ 𝑛 𝜏 𝜎 𝜔 {\displaystyle{\displaystyle\int_{-\tau}^{\tau}e^{-\mathrm{i}t\omega}\phi_{n}(% t)\mathrm{d}t=(-\mathrm{i})^{n}\sqrt{\frac{2\pi\tau\Lambda_{n}}{\sigma}}\phi_{% n}\left(\frac{\tau}{\sigma}\omega\right),}}
    0 references
    DLMF id
    DLMF:30.15.E5
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
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    Q12409
    Defining formula
    n ≥ m 𝑛 𝑚 {\displaystyle{\displaystyle n\geq m}}
    xml-id
    C30.S1.XMD5.m1ddec
    0 references
    Q12451
    Defining formula
    τ > 0 𝜏 0 {\displaystyle{\displaystyle\tau>0}}
    xml-id
    C30.S15.XMD1.m1cdec
    0 references
    Q12452
    Defining formula
    σ > 0 𝜎 0 {\displaystyle{\displaystyle\sigma>0}}
    xml-id
    C30.S15.XMD2.m1bdec
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    function (locally)
    Defining formula
    ϕ n ⁢ ( t ) subscript italic-ϕ 𝑛 𝑡 {\displaystyle{\displaystyle\phi_{n}(t)}}
    xml-id
    C30.S15.XMD3.m1cdec
    0 references
    DLMF:30.15.E2
    Defining formula
    Λ Λ {\displaystyle{\displaystyle\Lambda}}
    xml-id
    C30.S15.XMD4.m1ddec
    0 references
     
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