DLMF:15.9.E16 (Q5106)

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DLMF:15.9.E16
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    Statements

    Defining formula
    𝐅 ⁑ ( a , b 2 ⁒ b ; z ) = Ο€ Ξ“ ⁑ ( b ) ⁒ z - b + ( 1 / 2 ) ⁒ ( 1 - z ) ( b - a - ( 1 / 2 ) ) / 2 ⁒ P a - b - ( 1 / 2 ) - b + ( 1 / 2 ) ⁑ ( 2 - z 2 ⁒ 1 - z ) , scaled-hypergeometric-bold-F π‘Ž 𝑏 2 𝑏 𝑧 πœ‹ Euler-Gamma 𝑏 superscript 𝑧 𝑏 1 2 superscript 1 𝑧 𝑏 π‘Ž 1 2 2 Legendre-P-first-kind 𝑏 1 2 π‘Ž 𝑏 1 2 2 𝑧 2 1 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop 2b};z\right)=\frac{\sqrt% {\pi}}{\Gamma\left(b\right)}z^{-b+(\ifrac{1}{2})}(1-z)^{(b-a-(\ifrac{1}{2}))/2% }\*P^{-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(\frac{2-z}{2\sqrt{1-z}}% \right),}}
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    DLMF id
    DLMF:15.9.E16
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    constraint
    | ph ⁑ ( 1 - z ) | < Ο€ phase 1 𝑧 πœ‹ {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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    | ph ⁑ ( 1 - z ) | < Ο€ ph 1 𝑧 πœ‹ {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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    | 1 - z | < 1 1 𝑧 1 {\displaystyle{\displaystyle|1-z|<1}}
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    Symbols used
    gamma function
    Defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
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    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.m1adec
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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.m2aadec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
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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.m2adec
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