DLMF:9.12.E29 (Q2957)

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DLMF:9.12.E29
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    Statements

    Defining formula
    Hi ⁡ ( z ) ∼ - 1 π ⁢ z ⁢ ∑ k = 0 ∞ ( 3 ⁢ k ) ! k ! ⁢ ( 3 ⁢ z 3 ) k + e ζ π ⁢ z 1 / 4 ⁢ ∑ k = 0 ∞ u k ζ k , asymptotic-expansion Scorer-Hi 𝑧 1 𝜋 𝑧 superscript subscript 𝑘 0 3 𝑘 𝑘 superscript 3 superscript 𝑧 3 𝑘 superscript 𝑒 𝜁 𝜋 superscript 𝑧 1 4 superscript subscript 𝑘 0 subscript 𝑢 𝑘 superscript 𝜁 𝑘 {\displaystyle{\displaystyle\mathrm{Hi}\left(z\right)\sim-\frac{1}{\pi z}\sum_% {k=0}^{\infty}\frac{(3k)!}{k!(3z^{3})^{k}}+\frac{e^{\zeta}}{\sqrt{\pi}z^{1/4}}% \sum_{k=0}^{\infty}\frac{u_{k}}{\zeta^{k}},}}
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    DLMF id
    DLMF:9.12.E29
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    constraint
    | ph ⁡ z | ≤ π - δ phase 𝑧 𝜋 𝛿 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\pi-\delta}}
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