DLMF:17.2.E48 (Q5338)

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DLMF:17.2.E48
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    Statements

    Defining formula
    ∫ 0 ∞ f ⁢ ( x ) ⁢ d q x = lim n → ∞ ⁡ ∫ 0 q - n f ⁢ ( x ) ⁢ d q x = ( 1 - q ) ⁢ ∑ j = - ∞ ∞ f ⁢ ( q j ) ⁢ q j , superscript subscript 0 𝑓 𝑥 q-differential 𝑞 𝑥 subscript → 𝑛 superscript subscript 0 superscript 𝑞 𝑛 𝑓 𝑥 q-differential 𝑞 𝑥 1 𝑞 superscript subscript 𝑗 𝑓 superscript 𝑞 𝑗 superscript 𝑞 𝑗 {\displaystyle{\displaystyle\int_{0}^{\infty}f(x){\mathrm{d}}_{q}x=\lim_{n\to% \infty}\int_{0}^{q^{-n}}f(x){\mathrm{d}}_{q}x=(1-q)\sum_{j=-\infty}^{\infty}f(% q^{j})q^{j},}}
    0 references
    DLMF id
    DLMF:17.2.E48
    0 references
    Symbols used
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
    0 references
    Q11299
    Defining formula
    d q x q-differential 𝑞 𝑥 {\displaystyle{\displaystyle{\mathrm{d}}_{\NVar{q}}\NVar{x}}}
    xml-id
    C17.S2.SS5.p1.m3acdec
    0 references
    Q11710
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C17.S1.XMD10.m1atdec
    0 references
    Q11712
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C17.S1.XMD2.m1mdec
    0 references
    Q11711
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C17.S1.XMD4.m1amdec
    0 references
    Q11719
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C17.S1.XMD8.m1cdec
    0 references
     
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