DLMF:12.5.E6 (Q4120): 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.m1aedec

Revision as of 15:10, 2 January 2020

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DLMF:12.5.E6
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    Statements

    Defining formula
    U ⁡ ( a , z ) = e 1 4 ⁢ z 2 i ⁢ 2 ⁢ π ⁢ ∫ c - i ⁢ ∞ c + i ⁢ ∞ e - z ⁢ t + 1 2 ⁢ t 2 ⁢ t - a - 1 2 ⁢ d t , parabolic-U 𝑎 𝑧 superscript 𝑒 1 4 superscript 𝑧 2 𝑖 2 𝜋 superscript subscript 𝑐 𝑖 𝑐 𝑖 superscript 𝑒 𝑧 𝑡 1 2 superscript 𝑡 2 superscript 𝑡 𝑎 1 2 𝑡 {\displaystyle{\displaystyle U\left(a,z\right)=\frac{e^{\frac{1}{4}z^{2}}}{i% \sqrt{2\pi}}\int_{c-i\infty}^{c+i\infty}e^{-zt+\frac{1}{2}t^{2}}t^{-a-\frac{1}% {2}}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:12.5.E6
    0 references
    constraint
    - 1 2 ⁢ π < ph ⁡ t < 1 2 ⁢ π 1 2 𝜋 phase 𝑡 1 2 𝜋 {\displaystyle{\displaystyle-\tfrac{1}{2}\pi<\operatorname{ph}t<\tfrac{1}{2}% \pi}}
    0 references
    - 1 2 ⁢ π < ph ⁡ t < 1 2 ⁢ π 1 2 𝜋 ph 𝑡 1 2 𝜋 {\displaystyle{\displaystyle-\tfrac{1}{2}\pi<\operatorname{ph}t<\tfrac{1}{2}% \pi}}
    0 references
    c > 0 𝑐 0 {\displaystyle{\displaystyle c>0}}
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    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.m1aedec
    0 references
     
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