DLMF:11.9.E5 (Q4012)

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DLMF:11.9.E5
No description defined

    Statements

    Defining formula
    S μ , ν ⁡ ( z ) = s μ , ν ⁡ ( z ) + 2 μ - 1 ⁢ Γ ⁡ ( 1 2 ⁢ μ + 1 2 ⁢ ν + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν + 1 2 ) ⁢ ( sin ⁡ ( 1 2 ⁢ ( μ - ν ) ⁢ π ) ⁢ J ν ⁡ ( z ) - cos ⁡ ( 1 2 ⁢ ( μ - ν ) ⁢ π ) ⁢ Y ν ⁡ ( z ) ) , Lommel-S 𝜇 𝜈 𝑧 Lommel-s 𝜇 𝜈 𝑧 superscript 2 𝜇 1 Euler-Gamma 1 2 𝜇 1 2 𝜈 1 2 Euler-Gamma 1 2 𝜇 1 2 𝜈 1 2 1 2 𝜇 𝜈 𝜋 Bessel-J 𝜈 𝑧 1 2 𝜇 𝜈 𝜋 Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle S_{{\mu},{\nu}}\left(z\right)=s_{{\mu},{\nu}}% \left(z\right)+2^{\mu-1}\Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{% 2}\right)\Gamma\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\*% \left(\sin\left(\tfrac{1}{2}(\mu-\nu)\pi\right)\,J_{\nu}\left(z\right)-\cos% \left(\tfrac{1}{2}(\mu-\nu)\pi\right)\,Y_{\nu}\left(z\right)\right),}}
    0 references
    DLMF id
    DLMF:11.9.E5
    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
    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.m2aadec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Lommel function
    Defining formula
    s μ , ν ⁡ ( z ) Lommel-s 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle s_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right% )}}
    xml-id
    C11.S9.E3.m2abdec
    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
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q11491
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C11.S1.XMD2.m1cdec
    0 references
    Q11492
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C11.S1.XMD3.m1ddec
    0 references
     
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