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DLMF:10.22#Ex4 - MaRDI portal

DLMF:10.22#Ex4 (Q4073)

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DLMF:10.22#Ex4
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    e i z z - ν 𝒞 ν ( z ) d z = e i z z - ν + 1 1 - 2 ν ( 𝒞 ν ( z ) + i 𝒞 ν - 1 ( z ) ) , superscript 𝑒 𝑖 𝑧 superscript 𝑧 𝜈 cylinder-function 𝜈 𝑧 𝑧 superscript 𝑒 𝑖 𝑧 superscript 𝑧 𝜈 1 1 2 𝜈 cylinder-function 𝜈 𝑧 𝑖 cylinder-function 𝜈 1 𝑧 {\displaystyle{\displaystyle\int e^{iz}z^{-\nu}\mathscr{C}_{\nu}\left(z\right)% \mathrm{d}z=\frac{e^{iz}z^{-\nu+1}}{1-2\nu}(\mathscr{C}_{\nu}\left(z\right)+i% \mathscr{C}_{\nu-1}\left(z\right)),}}
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    ν 1 2 𝜈 1 2 {\displaystyle{\displaystyle\nu\neq\tfrac{1}{2}}}
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