DLMF:25.14.E6 (Q7745): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aadec

Revision as of 01:03, 2 January 2020

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DLMF:25.14.E6
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    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,}}
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    DLMF id
    DLMF:25.14.E6
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    constraint
    ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}}
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    ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}}
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    | z | = 1 , ℜ ⁡ a > 0 formulae-sequence 𝑧 1 𝑎 0 {\displaystyle{\displaystyle|z|=1,\Re a>0}}
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    ℜ ⁡ 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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    ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}}
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    | z | = 1 , ℜ ⁡ a > 0 formulae-sequence 𝑧 1 𝑎 0 {\displaystyle{\displaystyle|z|=1,\Re a>0}}
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    Symbols used
    Lerch’s transcendent
    Defining formula
    Φ ⁡ ( z , s , a ) Lerch-Phi 𝑧 𝑠 𝑎 {\displaystyle{\displaystyle\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S14.E1.m2aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
     
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