DLMF:10.22.E6 (Q3379)

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DLMF:10.22.E6
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    ∫ 𝒞 μ ⁡ ( a ⁢ z ) ⁢ 𝒟 ν ⁢ ( a ⁢ z ) ⁢ d z z = - a ⁢ z ⁢ ( 𝒞 μ + 1 ⁡ ( a ⁢ z ) ⁢ 𝒟 ν ⁢ ( a ⁢ z ) - 𝒞 μ ⁡ ( a ⁢ z ) ⁢ 𝒟 ν + 1 ⁢ ( a ⁢ z ) ) μ 2 - ν 2 + 𝒞 μ ⁡ ( a ⁢ z ) ⁢ 𝒟 ν ⁢ ( a ⁢ z ) μ + ν , cylinder-function 𝜇 𝑎 𝑧 subscript 𝒟 𝜈 𝑎 𝑧 𝑧 𝑧 𝑎 𝑧 cylinder-function 𝜇 1 𝑎 𝑧 subscript 𝒟 𝜈 𝑎 𝑧 cylinder-function 𝜇 𝑎 𝑧 subscript 𝒟 𝜈 1 𝑎 𝑧 superscript 𝜇 2 superscript 𝜈 2 cylinder-function 𝜇 𝑎 𝑧 subscript 𝒟 𝜈 𝑎 𝑧 𝜇 𝜈 {\displaystyle{\displaystyle\int\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{% \nu}(az)\frac{\mathrm{d}z}{z}=-\frac{az(\mathscr{C}_{\mu+1}\left(az\right)% \mathscr{D}_{\nu}(az)-\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\nu+1}(az))% }{\mu^{2}-\nu^{2}}+\frac{\mathscr{C}_{\mu}\left(az\right)\mathscr{D}_{\nu}(az)% }{\mu+\nu},}}
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    DLMF:10.22.E6
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