DLMF:10.40.E5 (Q3583)

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DLMF:10.40.E5
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    test
    I ν ⁡ ( z ) ∼ e z ( 2 ⁢ π ⁢ z ) 1 2 ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ a k ⁢ ( ν ) z k ± i ⁢ e ± ν ⁢ π ⁢ i ⁢ e - z ( 2 ⁢ π ⁢ z ) 1 2 ⁢ ∑ k = 0 ∞ a k ⁢ ( ν ) z k , asymptotic-expansion modified-Bessel-first-kind 𝜈 𝑧 plus-or-minus superscript 𝑒 𝑧 superscript 2 𝜋 𝑧 1 2 superscript subscript 𝑘 0 superscript 1 𝑘 subscript 𝑎 𝑘 𝜈 superscript 𝑧 𝑘 𝑖 superscript 𝑒 plus-or-minus 𝜈 𝜋 𝑖 superscript 𝑒 𝑧 superscript 2 𝜋 𝑧 1 2 superscript subscript 𝑘 0 subscript 𝑎 𝑘 𝜈 superscript 𝑧 𝑘 {\displaystyle{\displaystyle I_{\nu}\left(z\right)\sim\frac{e^{z}}{(2\pi z)^{% \frac{1}{2}}}\sum_{k=0}^{\infty}(-1)^{k}\frac{a_{k}(\nu)}{z^{k}}\pm ie^{\pm\nu% \pi i}\frac{e^{-z}}{(2\pi z)^{\frac{1}{2}}}\sum_{k=0}^{\infty}\frac{a_{k}(\nu)% }{z^{k}},}}
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    DLMF:10.40.E5
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