DLMF:12.9.E4 (Q4153): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
Defining formula:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2aadec

Revision as of 15:16, 2 January 2020

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DLMF:12.9.E4
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    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
     
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