DLMF:30.15.E4 (Q8946)

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DLMF:30.15.E4
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    Statements

    Defining formula
    ∫ - ∞ ∞ e - i ⁢ t ⁢ ω ⁢ ϕ n ⁢ ( t ) ⁢ d t = ( - i ) n ⁢ 2 ⁢ π ⁢ τ σ ⁢ Λ n ⁢ ϕ n ⁢ ( τ σ ⁢ ω ) ⁢ χ σ ⁢ ( ω ) , superscript subscript superscript 𝑒 imaginary-unit 𝑡 𝜔 subscript italic-ϕ 𝑛 𝑡 𝑡 superscript imaginary-unit 𝑛 2 𝜋 𝜏 𝜎 subscript Λ 𝑛 subscript italic-ϕ 𝑛 𝜏 𝜎 𝜔 subscript 𝜒 𝜎 𝜔 {\displaystyle{\displaystyle\int_{-\infty}^{\infty}e^{-\mathrm{i}t\omega}\phi_% {n}(t)\mathrm{d}t=(-\mathrm{i})^{n}\sqrt{\frac{2\pi\tau}{\sigma\Lambda_{n}}}% \phi_{n}\left(\frac{\tau}{\sigma}\omega\right)\chi_{\sigma}(\omega),}}
    0 references
    DLMF id
    DLMF:30.15.E4
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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.m2abdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
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    Q12409
    Defining formula
    n ≥ m 𝑛 𝑚 {\displaystyle{\displaystyle n\geq m}}
    xml-id
    C30.S1.XMD5.m1cdec
    0 references
    Q12451
    Defining formula
    τ > 0 𝜏 0 {\displaystyle{\displaystyle\tau>0}}
    xml-id
    C30.S15.XMD1.m1bdec
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