DLMF:12.9.E3 (Q4152)

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DLMF:12.9.E3
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    Statements

    Defining formula
    U ⁡ ( a , z ) ∼ e - 1 4 ⁢ z 2 ⁢ z - a - 1 2 ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ ( 1 2 + a ) 2 ⁢ s s ! ⁢ ( 2 ⁢ z 2 ) s ± i ⁢ 2 ⁢ π Γ ⁡ ( 1 2 + a ) ⁢ e ∓ i ⁢ π ⁢ a ⁢ e 1 4 ⁢ z 2 ⁢ z a - 1 2 ⁢ ∑ s = 0 ∞ ( 1 2 - a ) 2 ⁢ s s ! ⁢ ( 2 ⁢ z 2 ) s , asymptotic-expansion parabolic-U 𝑎 𝑧 plus-or-minus superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑧 𝑎 1 2 superscript subscript 𝑠 0 superscript 1 𝑠 Pochhammer 1 2 𝑎 2 𝑠 𝑠 superscript 2 superscript 𝑧 2 𝑠 𝑖 2 𝜋 Euler-Gamma 1 2 𝑎 superscript 𝑒 minus-or-plus 𝑖 𝜋 𝑎 superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑧 𝑎 1 2 superscript subscript 𝑠 0 Pochhammer 1 2 𝑎 2 𝑠 𝑠 superscript 2 superscript 𝑧 2 𝑠 {\displaystyle{\displaystyle U\left(a,z\right)\sim e^{-\frac{1}{4}z^{2}}z^{-a-% \frac{1}{2}}\sum_{s=0}^{\infty}(-1)^{s}\frac{{\left(\frac{1}{2}+a\right)_{2s}}% }{s!(2z^{2})^{s}}\pm i\frac{\sqrt{2\pi}}{\Gamma\left(\tfrac{1}{2}+a\right)}e^{% \mp i\pi a}e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}\sum_{s=0}^{\infty}\frac{{% \left(\tfrac{1}{2}-a\right)_{2s}}}{s!(2z^{2})^{s}},}}
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    DLMF id
    DLMF:12.9.E3
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    constraint
    1 4 ⁢ π + δ ≤ ± ph ⁡ z ≤ 5 4 ⁢ π - δ 1 4 𝜋 𝛿 plus-or-minus phase 𝑧 5 4 𝜋 𝛿 {\displaystyle{\displaystyle\tfrac{1}{4}\pi+\delta\leq\pm\operatorname{ph}z% \leq\tfrac{5}{4}\pi-\delta}}
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