DLMF:25.14.E6 (Q7745)

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DLMF:25.14.E6
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    Statements

    test
    Ξ¦ ⁑ ( z , s , a ) = 1 2 ⁒ a - s + ∫ 0 ∞ z x ( a + x ) s ⁒ d x - 2 ⁒ ∫ 0 ∞ sin ⁑ ( x ⁒ ln ⁑ z - s ⁒ arctan ⁑ ( x / a ) ) ( a 2 + x 2 ) s / 2 ⁒ ( e 2 ⁒ Ο€ ⁒ x - 1 ) ⁒ d x , Lerch-Phi 𝑧 𝑠 π‘Ž 1 2 superscript π‘Ž 𝑠 superscript subscript 0 superscript 𝑧 π‘₯ superscript π‘Ž π‘₯ 𝑠 π‘₯ 2 superscript subscript 0 π‘₯ 𝑧 𝑠 π‘₯ π‘Ž superscript superscript π‘Ž 2 superscript π‘₯ 2 𝑠 2 superscript 𝑒 2 πœ‹ π‘₯ 1 π‘₯ {\displaystyle{\displaystyle\Phi\left(z,s,a\right)=\frac{1}{2}a^{-s}+\int_{0}^% {\infty}\frac{z^{x}}{(a+x)^{s}}\mathrm{d}x-2\int_{0}^{\infty}\frac{\sin\left(x% \ln z-s\operatorname{arctan}\left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}(e^{2% \pi x}-1)}\mathrm{d}x,}}
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    DLMF id
    DLMF:25.14.E6
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    constraint
    β„œ ⁑ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}}
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    β„œ ⁑ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}}
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    | z | = 1 , β„œ ⁑ a > 0 formulae-sequence 𝑧 1 π‘Ž 0 {\displaystyle{\displaystyle|z|=1,\Re a>0}}
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    β„œ ⁑ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}}
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    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
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    β„œ ⁑ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}}
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    | z | = 1 , β„œ ⁑ a > 0 formulae-sequence 𝑧 1 π‘Ž 0 {\displaystyle{\displaystyle|z|=1,\Re a>0}}
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    Symbols used
    Lerch’s transcendent
    test
    Ξ¦ ⁑ ( z , s , a ) Lerch-Phi 𝑧 𝑠 π‘Ž {\displaystyle{\displaystyle\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S14.E1.m2aedec
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    the ratio of the circumference of a circle to its diameter
    test
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
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    Q10770
    test
    d x π‘₯ {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
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    Q10816
    test
    arctan ⁑ z 𝑧 {\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m7adec
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