DLMF:10.22.E21 (Q3395)

From DRMF
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:10.22.E21
No description defined

    Statements

    Defining formula
    ∫ 0 1 2 ⁢ π Y μ ⁡ ( z ⁢ sin ⁡ θ ) ⁢ ( sin ⁡ θ ) μ ⁢ ( cos ⁡ θ ) 2 ⁢ μ ⁢ d θ = π 1 2 ⁢ 2 μ - 1 ⁢ z - μ ⁢ Γ ⁡ ( μ + 1 2 ) ⁢ J μ ⁡ ( 1 2 ⁢ z ) ⁢ Y μ ⁡ ( 1 2 ⁢ z ) , superscript subscript 0 1 2 𝜋 Bessel-Y-Weber 𝜇 𝑧 𝜃 superscript 𝜃 𝜇 superscript 𝜃 2 𝜇 𝜃 superscript 𝜋 1 2 superscript 2 𝜇 1 superscript 𝑧 𝜇 Euler-Gamma 𝜇 1 2 Bessel-J 𝜇 1 2 𝑧 Bessel-Y-Weber 𝜇 1 2 𝑧 {\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}Y_{\mu}\left(z\sin\theta% \right)(\sin\theta)^{\mu}(\cos\theta)^{2\mu}\mathrm{d}\theta=\pi^{\frac{1}{2}}% 2^{\mu-1}z^{-\mu}\*\Gamma\left(\mu+\tfrac{1}{2}\right)J_{\mu}\left(\tfrac{1}{2% }z\right)Y_{\mu}\left(\tfrac{1}{2}z\right),}}
    0 references
    DLMF id
    DLMF:10.22.E21
    0 references
    constraint
    ℜ ⁡ μ > - 1 2 𝜇 1 2 {\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2}}}
    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.m2amdec
    0 references
    Bessel function of the second kind
    Defining formula
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2abdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aidec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2agdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1atdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3atdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1ahdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2afdec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1odec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q3395&oldid=30068"