DLMF:12.2.E19 (Q4104)

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DLMF:12.2.E19
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    Statements

    Defining formula
    U ⁡ ( a , z ) = ± i ⁢ e ± i ⁢ π ⁢ a ⁢ U ⁡ ( a , - z ) + 2 ⁢ π Γ ⁡ ( 1 2 + a ) ⁢ e ± i ⁢ π ⁢ ( 1 2 ⁢ a - 1 4 ) ⁢ U ⁡ ( - a , ± i ⁢ z ) . parabolic-U 𝑎 𝑧 plus-or-minus 𝑖 superscript 𝑒 plus-or-minus 𝑖 𝜋 𝑎 parabolic-U 𝑎 𝑧 2 𝜋 Euler-Gamma 1 2 𝑎 superscript 𝑒 plus-or-minus 𝑖 𝜋 1 2 𝑎 1 4 parabolic-U 𝑎 plus-or-minus 𝑖 𝑧 {\displaystyle{\displaystyle U\left(a,z\right)=\pm ie^{\pm i\pi a}U\left(a,-z% \right)+\frac{\sqrt{2\pi}}{\Gamma\left(\tfrac{1}{2}+a\right)}e^{\pm i\pi(\frac% {1}{2}a-\frac{1}{4})}U\left(-a,\pm iz\right).}}
    0 references
    DLMF id
    DLMF:12.2.E19
    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.m2aidec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2akdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    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.m1akdec
    0 references
    Q11513
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C12.S1.XMD3.m1ndec
    0 references
    Q11516
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C12.S1.XMD6.m1kdec
    0 references
     
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