DLMF:31.7.E4 (Q9027)

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DLMF:31.7.E4
No description defined

    Statements

    Defining formula
    H ⁢ ℓ ⁡ ( 1 2 + i ⁢ 3 2 , α ⁢ β ⁢ ( 1 2 + i ⁢ 3 6 ) ; α , β , 1 3 ⁢ ( α + β + 1 ) , 1 3 ⁢ ( α + β + 1 ) ; z ) = F 1 2 ⁡ ( 1 3 ⁢ α , 1 3 ⁢ β ; 1 3 ⁢ ( α + β + 1 ) ; 1 - ( 1 - ( 3 2 - i ⁢ 3 2 ) ⁢ z ) 3 ) . Heun-local 1 2 𝑖 3 2 𝛼 𝛽 1 2 𝑖 3 6 𝛼 𝛽 1 3 𝛼 𝛽 1 1 3 𝛼 𝛽 1 𝑧 Gauss-hypergeometric-F-as-2F1 1 3 𝛼 1 3 𝛽 1 3 𝛼 𝛽 1 1 superscript 1 3 2 𝑖 3 2 𝑧 3 {\displaystyle{\displaystyle\mathit{H\!\ell}\left(\tfrac{1}{2}+i\tfrac{\sqrt{3% }}{2},\alpha\beta(\tfrac{1}{2}+i\tfrac{\sqrt{3}}{6});\alpha,\beta,\tfrac{1}{3}% (\alpha+\beta+1),\tfrac{1}{3}(\alpha+\beta+1);z\right)={{}_{2}F_{1}}\left(% \tfrac{1}{3}\alpha,\tfrac{1}{3}\beta;\tfrac{1}{3}(\alpha+\beta+1);1-\left(1-% \left(\tfrac{3}{2}-i\tfrac{\sqrt{3}}{2}\right)z\right)^{3}\right).}}
    0 references
    DLMF id
    DLMF:31.7.E4
    0 references
    Symbols used
    Q11581
    Defining formula
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5acdec
    0 references
    Heun functions
    Defining formula
    H ⁢ ℓ ⁡ ( a , q ; α , β , γ , missing ; missing ) Heun-local 𝑎 𝑞 𝛼 𝛽 𝛾 missing missing {\displaystyle{\displaystyle\mathit{H\!\ell}\left(\NVar{a},\NVar{q};\NVar{% \alpha},\NVar{\beta},\NVar{\gamma},\NVar{missing};missing\right)\)\@add@PDF@RDFa@triples\end{document}}}
    xml-id
    C31.S3.E1.m2acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q12457
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C31.S1.XMD1.m1cdec
    0 references
    Q12463
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C31.S1.XMD8.m1cdec
    0 references
    Q12464
    Defining formula
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C31.S1.XMD9.m1cdec
    0 references
     
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