DLMF:12.2.E20 (Q4105)

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DLMF:12.2.E20
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    Statements

    Defining formula
    V ⁡ ( a , z ) = ∓ i Γ ⁡ ( 1 2 - a ) ⁢ U ⁡ ( a , z ) + 2 π ⁢ e ∓ i ⁢ π ⁢ ( 1 2 ⁢ a - 1 4 ) ⁢ U ⁡ ( - a , ± i ⁢ z ) . parabolic-V 𝑎 𝑧 minus-or-plus 𝑖 Euler-Gamma 1 2 𝑎 parabolic-U 𝑎 𝑧 2 𝜋 superscript 𝑒 minus-or-plus 𝑖 𝜋 1 2 𝑎 1 4 parabolic-U 𝑎 plus-or-minus 𝑖 𝑧 {\displaystyle{\displaystyle V\left(a,z\right)=\frac{\mp i}{\Gamma\left(\frac{% 1}{2}-a\right)}U\left(a,z\right)+\sqrt{\frac{2}{\pi}}e^{\mp i\pi(\frac{1}{2}a-% \frac{1}{4})}U\left(-a,\pm iz\right).}}
    0 references
    DLMF id
    DLMF:12.2.E20
    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.m2ajdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aldec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
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    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.m1aldec
    0 references
    Q11517
    Defining formula
    V ⁡ ( a , z ) parabolic-V 𝑎 𝑧 {\displaystyle{\displaystyle V\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m2afdec
    0 references
    Q11513
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C12.S1.XMD3.m1odec
    0 references
     
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