DLMF:25.14.E6 (Q7745)

From DRMF
Revision as of 18:19, 30 December 2019 by Admin (talk | contribs) (‎Changed an Item: Add constraint)
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:25.14.E6
No description defined

    Statements

    Defining formula
    Φ ⁡ ( 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 ⁢ π ⁢ x - 1 ) ⁢ d x , Lerch-Phi 𝑧 𝑠 𝑎 1 2 superscript 𝑎 𝑠 superscript subscript 0 superscript 𝑧 𝑥 superscript 𝑎 𝑥 𝑠 𝑥 2 superscript subscript 0 𝑥 𝑧 𝑠 𝑥 𝑎 superscript superscript 𝑎 2 superscript 𝑥 2 𝑠 2 superscript 𝑒 2 𝜋 𝑥 1 𝑥 {\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,}}
    0 references
    DLMF id
    DLMF:25.14.E6
    0 references
    constraint
    ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}}
    0 references
    ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}}
    0 references
    | z | = 1 , ℜ ⁡ a > 0 formulae-sequence 𝑧 1 𝑎 0 {\displaystyle{\displaystyle|z|=1,\Re a>0}}
    0 references
    ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}}
    0 references
    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
    0 references
    ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}}
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q7745&oldid=19719"