DLMF:12.2.E18 (Q4103)

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DLMF:12.2.E18
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    Statements

    Defining formula
    2 ⁢ π ⁢ U ⁡ ( a , z ) = Γ ⁡ ( 1 2 - a ) ⁢ ( e ∓ i ⁢ π ⁢ ( 1 2 ⁢ a + 1 4 ) ⁢ U ⁡ ( - a , ± i ⁢ z ) + e ± i ⁢ π ⁢ ( 1 2 ⁢ a + 1 4 ) ⁢ U ⁡ ( - a , ∓ i ⁢ z ) ) , 2 𝜋 parabolic-U 𝑎 𝑧 Euler-Gamma 1 2 𝑎 superscript 𝑒 minus-or-plus 𝑖 𝜋 1 2 𝑎 1 4 parabolic-U 𝑎 plus-or-minus 𝑖 𝑧 superscript 𝑒 plus-or-minus 𝑖 𝜋 1 2 𝑎 1 4 parabolic-U 𝑎 minus-or-plus 𝑖 𝑧 {\displaystyle{\displaystyle\sqrt{2\pi}U\left(a,z\right)=\Gamma\left(\tfrac{1}% {2}-a\right)\left(e^{\mp i\pi(\frac{1}{2}a+\frac{1}{4})}U\left(-a,\pm iz\right% )+e^{\pm i\pi(\frac{1}{2}a+\frac{1}{4})}U\left(-a,\mp iz\right)\right),}}
    0 references
    DLMF id
    DLMF:12.2.E18
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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.m2ahdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ajdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
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