DLMF:10.22.E38 (Q3412)

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DLMF:10.22.E38
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    Statements

    Defining formula
    ∫ 0 1 t ⁢ J ν ⁡ ( α ℓ ⁢ t ) ⁢ J ν ⁡ ( α m ⁢ t ) ⁢ d t = ( a 2 b 2 + α ℓ 2 - ν 2 ) ⁢ ( J ν ⁡ ( α ℓ ) ) 2 2 ⁢ α ℓ 2 ⁢ δ ℓ , m , superscript subscript 0 1 𝑡 Bessel-J 𝜈 subscript 𝛼 ℓ 𝑡 Bessel-J 𝜈 subscript 𝛼 𝑚 𝑡 𝑡 superscript 𝑎 2 superscript 𝑏 2 superscript subscript 𝛼 ℓ 2 superscript 𝜈 2 superscript Bessel-J 𝜈 subscript 𝛼 ℓ 2 2 superscript subscript 𝛼 ℓ 2 Kronecker ℓ 𝑚 {\displaystyle{\displaystyle\int_{0}^{1}tJ_{\nu}\left(\alpha_{\ell}t\right)J_{% \nu}\left(\alpha_{m}t\right)\mathrm{d}t=\left(\frac{a^{2}}{b^{2}}+\alpha_{\ell% }^{2}-\nu^{2}\right)\frac{(J_{\nu}\left(\alpha_{\ell}\right))^{2}}{2\alpha_{% \ell}^{2}}\delta_{\ell,m},}}
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    DLMF id
    DLMF:10.22.E38
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