1.17.E18 (Q647)

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1.17.E18
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    Statements

    DLMF defining formula
    ∫ - π π ϕ ⁢ ( x ) ⁢ ( 1 2 ⁢ π ⁢ ∑ k = - ∞ ∞ e i ⁢ k ⁢ ( x - a ) ) ⁢ d x = ϕ ⁢ ( a ) . superscript subscript 𝜋 𝜋 italic-ϕ 𝑥 1 2 𝜋 superscript subscript 𝑘 superscript 𝑒 𝑖 𝑘 𝑥 𝑎 𝑥 italic-ϕ 𝑎 {\displaystyle{\displaystyle\int_{-\pi}^{\pi}\phi(x)\left(\frac{1}{2\pi}\sum_{% k=-\infty}^{\infty}e^{ik(x-a)}\right)\mathrm{d}x=\phi(a).}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ajdec
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    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1andec
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    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aidec
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    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2afdec
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    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3andec
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    integer
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C1.S1.XMD4.m1adec
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    continuous function
    DLMF defining formula
    ϕ ⁢ ( x ) italic-ϕ 𝑥 {\displaystyle{\displaystyle\phi(x)}}
    xml-id
    C1.S17.XMD1.m1gdec
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    Identifiers

     
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