DLMF:15.9.E13 (Q5103)

From DRMF
Revision as of 14:10, 2 January 2020 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:15.9.E13
No description defined

    Statements

    Defining formula
    f ⁢ ( t ) = 1 2 ⁢ π ⁢ i ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ f ~ ⁢ ( i ⁢ λ ) ⁢ Φ i ⁢ λ ( α , β ) ⁢ ( t ) ⁢ Γ ⁡ ( 1 2 ⁢ ( α + β + 1 + λ ) ) ⁢ Γ ⁡ ( 1 2 ⁢ ( α - β + 1 + λ ) ) Γ ⁡ ( α + 1 ) ⁢ Γ ⁡ ( λ ) ⁢ 2 α + β + 1 - λ ⁢ d λ , 𝑓 𝑡 1 2 𝜋 imaginary-unit superscript subscript imaginary-unit imaginary-unit ~ 𝑓 imaginary-unit 𝜆 subscript superscript Φ 𝛼 𝛽 imaginary-unit 𝜆 𝑡 Euler-Gamma 1 2 𝛼 𝛽 1 𝜆 Euler-Gamma 1 2 𝛼 𝛽 1 𝜆 Euler-Gamma 𝛼 1 Euler-Gamma 𝜆 superscript 2 𝛼 𝛽 1 𝜆 𝜆 {\displaystyle{\displaystyle f(t)=\frac{1}{2\pi\mathrm{i}}\int_{-\mathrm{i}% \infty}^{\mathrm{i}\infty}\widetilde{f}(\mathrm{i}\lambda)\Phi^{(\alpha,\beta)% }_{\mathrm{i}\lambda}(t)\frac{\Gamma\left(\tfrac{1}{2}(\alpha+\beta+1+\lambda)% \right)\Gamma\left(\tfrac{1}{2}(\alpha-\beta+1+\lambda)\right)}{\Gamma\left(% \alpha+1\right)\Gamma\left(\lambda\right)2^{\alpha+\beta+1-\lambda}}\mathrm{d}% \lambda,}}
    0 references
    DLMF id
    DLMF:15.9.E13
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    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
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    Q10777
    Defining formula
    ( a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}
    xml-id
    introduction.Sx4.p1.t1.r29.m6adec
    0 references
    Q11665
    Defining formula
    f ⁢ ( t ) 𝑓 𝑡 {\displaystyle{\displaystyle f(t)}}
    xml-id
    C15.S9.XMD1.m1adec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5103&oldid=44172"