DLMF:12.5.E2 (Q4116): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11320 / rank
 
Normal rank
Property / Symbols used: Q11320 / qualifier
 
Defining formula:

U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}

\paraU@{\NVar{a}}{\NVar{z}}
Property / Symbols used: Q11320 / qualifier
 
xml-id: C12.S2.SS1.p1.m1aadec

Revision as of 15:08, 2 January 2020

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DLMF:12.5.E2
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    Statements

    Defining formula
    U ⁡ ( a , z ) = z ⁢ e - 1 4 ⁢ z 2 Γ ⁡ ( 1 4 + 1 2 ⁢ a ) ⁢ ∫ 0 ∞ t 1 2 ⁢ a - 3 4 ⁢ e - t ⁢ ( z 2 + 2 ⁢ t ) - 1 2 ⁢ a - 3 4 ⁢ d t , parabolic-U 𝑎 𝑧 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 Euler-Gamma 1 4 1 2 𝑎 superscript subscript 0 superscript 𝑡 1 2 𝑎 3 4 superscript 𝑒 𝑡 superscript superscript 𝑧 2 2 𝑡 1 2 𝑎 3 4 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\frac{ze^{-\frac{1}{4}z^{2}}}{% \Gamma\left(\frac{1}{4}+\frac{1}{2}a\right)}\*\int_{0}^{\infty}t^{\frac{1}{2}a% -\frac{3}{4}}e^{-t}\left(z^{2}+2t\right)^{-\frac{1}{2}a-\frac{3}{4}}\mathrm{d}% t,}}
    0 references
    DLMF id
    DLMF:12.5.E2
    0 references
    constraint
    | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}}
    0 references
    ℜ ⁡ a > - 1 2 𝑎 1 2 {\displaystyle{\displaystyle\Re a>-\tfrac{1}{2}}}
    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
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    Q11320
    Defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1aadec
    0 references
     
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