DLMF:18.17.E18 (Q5759): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older editNewer edit →
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11743 / rank
 
Normal rank
Property / Symbols used: Q11743 / qualifier
 
Defining formula:

y 𝑦 {\displaystyle{\displaystyle y}}

y
Property / Symbols used: Q11743 / qualifier
 
xml-id: C18.S1.XMD1.m1mdec

Revision as of 15:48, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:18.17.E18
No description defined

    Statements

    Defining formula
    ∫ 0 1 ( 1 - x 2 ) λ - 1 2 ⁢ C 2 ⁢ n + 1 ( λ ) ⁡ ( x ) ⁢ sin ⁡ ( x ⁢ y ) ⁢ d x = ( - 1 ) n ⁢ π ⁢ Γ ⁡ ( 2 ⁢ n + 2 ⁢ λ + 1 ) ⁢ J 2 ⁢ n + λ + 1 ⁡ ( y ) ( 2 ⁢ n + 1 ) ! ⁢ Γ ⁡ ( λ ) ⁢ ( 2 ⁢ y ) λ . superscript subscript 0 1 superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 2 𝑛 1 𝑥 𝑥 𝑦 𝑥 superscript 1 𝑛 𝜋 Euler-Gamma 2 𝑛 2 𝜆 1 Bessel-J 2 𝑛 𝜆 1 𝑦 2 𝑛 1 Euler-Gamma 𝜆 superscript 2 𝑦 𝜆 {\displaystyle{\displaystyle\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(% \lambda)}_{2n+1}\left(x\right)\sin\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\pi% \Gamma\left(2n+2\lambda+1\right)J_{2n+\lambda+1}\left(y\right)}{(2n+1)!\Gamma% \left(\lambda\right)(2y)^{\lambda}}.}}
    0 references
    DLMF id
    DLMF:18.17.E18
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aadec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aidec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aqdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aqdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
    0 references
    Q11475
    Defining formula
    C n ( λ ) ⁡ ( x ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 {\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}% \right)}}
    xml-id
    C18.S3.T1.t1.r3.m2addec
    0 references
    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1mdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5759&oldid=61065"