DLMF:15.9.E22 (Q5112): Difference between revisions

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Property / constraint
 

0 < | ph ⁡ z | < π 0 phase 𝑧 𝜋 {\displaystyle{\displaystyle 0<|\operatorname{ph}z|<\pi}}

0<|\phase@@{z}|<\pi
Property / constraint: 0 < | ph ⁡ z | < π 0 phase 𝑧 𝜋 {\displaystyle{\displaystyle 0<|\operatorname{ph}z|<\pi}} / rank
 
Normal rank

Revision as of 16:57, 30 December 2019

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DLMF:15.9.E22
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    Statements

    test
    𝐅 ⁡ ( a , b 1 2 ; z ) = 2 a + b - ( 3 / 2 ) π ⁢ Γ ⁡ ( a + 1 2 ) ⁢ Γ ⁡ ( b + 1 2 ) ⁢ ( z - 1 ) ( - a - b + ( 1 / 2 ) ) / 2 ⁢ ( e ± π ⁢ i ⁢ ( a + b - ( 1 / 2 ) ) ⁢ P a - b - ( 1 / 2 ) - a - b + ( 1 / 2 ) ⁡ ( - z ) + P a - b - ( 1 / 2 ) - a - b + ( 1 / 2 ) ⁡ ( z ) ) , scaled-hypergeometric-bold-F 𝑎 𝑏 1 2 𝑧 superscript 2 𝑎 𝑏 3 2 𝜋 Euler-Gamma 𝑎 1 2 Euler-Gamma 𝑏 1 2 superscript 𝑧 1 𝑎 𝑏 1 2 2 superscript 𝑒 plus-or-minus 𝜋 imaginary-unit 𝑎 𝑏 1 2 Legendre-P-first-kind 𝑎 𝑏 1 2 𝑎 𝑏 1 2 𝑧 Legendre-P-first-kind 𝑎 𝑏 1 2 𝑎 𝑏 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{1}{2}};z\right)=% \frac{2^{a+b-(\ifrac{3}{2})}}{\pi}\Gamma\left(a+\tfrac{1}{2}\right)\Gamma\left% (b+\tfrac{1}{2}\right)\*(z-1)^{(-a-b+(\ifrac{1}{2}))/2}\*\left(e^{\pm\pi% \mathrm{i}(a+b-(\ifrac{1}{2}))}P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}% \left(-\sqrt{z}\right)+P^{-a-b+(\ifrac{1}{2})}_{a-b-(\ifrac{1}{2})}\left(\sqrt% {z}\right)\right),}}
    0 references
    DLMF id
    DLMF:15.9.E22
    0 references
    constraint
    0 < | ph ⁡ z | < π 0 phase 𝑧 𝜋 {\displaystyle{\displaystyle 0<|\operatorname{ph}z|<\pi}}
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