DLMF:12.5.E8 (Q4122): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2agdec

Revision as of 15:10, 2 January 2020

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DLMF:12.5.E8
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    Statements

    Defining formula
    U ⁡ ( a , z ) = e - 1 4 ⁢ z 2 ⁢ z - a - 1 2 2 ⁢ π ⁢ i ⁢ Γ ⁡ ( 1 2 + a ) ⁢ ∫ - i ⁢ ∞ i ⁢ ∞ Γ ⁡ ( t ) ⁢ Γ ⁡ ( 1 2 + a - 2 ⁢ t ) ⁢ 2 t ⁢ z 2 ⁢ t ⁢ d t , parabolic-U 𝑎 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑧 𝑎 1 2 2 𝜋 𝑖 Euler-Gamma 1 2 𝑎 superscript subscript 𝑖 𝑖 Euler-Gamma 𝑡 Euler-Gamma 1 2 𝑎 2 𝑡 superscript 2 𝑡 superscript 𝑧 2 𝑡 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}z^{-% a-\frac{1}{2}}}{2\pi i\Gamma\left(\frac{1}{2}+a\right)}\*\int_{-i\infty}^{i% \infty}\Gamma\left(t\right)\Gamma\left(\tfrac{1}{2}+a-2t\right)2^{t}z^{2t}% \mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:12.5.E8
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    constraint
    | ph ⁡ z | < 3 4 ⁢ π phase 𝑧 3 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{3}{4}\pi}}
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    a ≠ - 1 2 , - 3 2 , - 5 2 , … 𝑎 1 2 3 2 5 2 … {\displaystyle{\displaystyle a\neq-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\dots}}
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    | ph ⁡ z | < 3 4 ⁢ π ph 𝑧 3 4 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{3}{4}\pi}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1agdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
    0 references
     
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