DLMF:10.41.E13 (Q3610)

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DLMF:10.41.E13
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    Statements

    Defining formula
    K ν ⁡ ( ν ⁢ z ) = ( π 2 ⁢ ν ) 1 2 ⁢ e - ν ⁢ η ( 1 + z 2 ) 1 4 ⁢ ( ∑ k = 0 ℓ - 1 ( - 1 ) k ⁢ U k ⁢ ( p ) ν k + O ⁡ ( 1 z ℓ ) ) , modified-Bessel-second-kind 𝜈 𝜈 𝑧 superscript 𝜋 2 𝜈 1 2 superscript 𝑒 𝜈 𝜂 superscript 1 superscript 𝑧 2 1 4 superscript subscript 𝑘 0 ℓ 1 superscript 1 𝑘 subscript 𝑈 𝑘 𝑝 superscript 𝜈 𝑘 Big-O 1 superscript 𝑧 ℓ {\displaystyle{\displaystyle K_{\nu}\left(\nu z\right)=\left(\frac{\pi}{2\nu}% \right)^{\frac{1}{2}}\frac{e^{-\nu\eta}}{(1+z^{2})^{\frac{1}{4}}}\*\left(\sum_% {k=0}^{\ell-1}(-1)^{k}\frac{U_{k}(p)}{\nu^{k}}+O\left(\frac{1}{z^{\ell}}\right% )\right),}}
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    DLMF id
    DLMF:10.41.E13
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    constraint
    | ph ⁡ z | ≤ 3 2 ⁢ π - δ phase 𝑧 3 2 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta}}
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