DLMF:15.9.E18 (Q5108)

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DLMF:15.9.E18
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    Statements

    Defining formula
    𝐅 ⁑ ( a , b a + b + 1 2 ; z ) = 2 a + b - ( 1 / 2 ) ⁒ ( - z ) ( - a - b + ( 1 / 2 ) ) / 2 ⁒ P a - b - ( 1 / 2 ) - a - b + ( 1 / 2 ) ⁑ ( 1 - z ) , scaled-hypergeometric-bold-F π‘Ž 𝑏 π‘Ž 𝑏 1 2 𝑧 superscript 2 π‘Ž 𝑏 1 2 superscript 𝑧 π‘Ž 𝑏 1 2 2 Legendre-P-first-kind π‘Ž 𝑏 1 2 π‘Ž 𝑏 1 2 1 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop a+b+\tfrac{1}{2}};z% \right)=2^{a+b-(\ifrac{1}{2})}(-z)^{(-a-b+(\ifrac{1}{2}))/2}\*P^{-a-b+(\ifrac{% 1}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt{1-z}\right),}}
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    DLMF id
    DLMF:15.9.E18
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    constraint
    | ph ⁑ ( - z ) | < Ο€ phase 𝑧 πœ‹ {\displaystyle{\displaystyle\left|\operatorname{ph}\left(-z\right)\right|<\pi}}
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    Symbols used
    Q11481
    Defining formula
    P Ξ½ ΞΌ ⁑ ( z ) Legendre-P-first-kind πœ‡ 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1abdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1abdec
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    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁑ ( a , b ; c ; z ) scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2abdec
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