DLMF:12.9.E4 (Q4153)

From DRMF
Revision as of 15:16, 2 January 2020 by Admin (talk | contribs) (‎Changed an Item: Add constraint)
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:12.9.E4
No description defined

    Statements

    Defining formula
    V ⁡ ( a , z ) ∼ 2 π ⁢ e 1 4 ⁢ z 2 ⁢ z a - 1 2 ⁢ ∑ s = 0 ∞ ( 1 2 - a ) 2 ⁢ s s ! ⁢ ( 2 ⁢ z 2 ) s ± i Γ ⁡ ( 1 2 - a ) ⁢ e - 1 4 ⁢ z 2 ⁢ z - a - 1 2 ⁢ ∑ s = 0 ∞ ( - 1 ) s ⁢ ( 1 2 + a ) 2 ⁢ s s ! ⁢ ( 2 ⁢ z 2 ) s , asymptotic-expansion parabolic-V 𝑎 𝑧 plus-or-minus 2 𝜋 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 𝑖 Euler-Gamma 1 2 𝑎 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 {\displaystyle{\displaystyle V\left(a,z\right)\sim\sqrt{\frac{2}{\pi}}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}}\pm\frac{i}{\Gamma\left(\tfrac{1}{2}-a\right)}e^% {-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}\sum_{s=0}^{\infty}(-1)^{s}\frac{{\left(% \tfrac{1}{2}+a\right)_{2s}}}{s!(2z^{2})^{s}},}}
    0 references
    DLMF id
    DLMF:12.9.E4
    0 references
    constraint
    - 1 4 ⁢ π + δ ≤ ± ph ⁡ z ≤ 3 4 ⁢ π - δ 1 4 𝜋 𝛿 plus-or-minus phase 𝑧 3 4 𝜋 𝛿 {\displaystyle{\displaystyle-\tfrac{1}{4}\pi+\delta\leq\pm\operatorname{ph}z% \leq\tfrac{3}{4}\pi-\delta}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1acdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q4153&oldid=56107"