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

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Property / Symbols used
 
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / rank
 
Normal rank
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
 
test:

𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}

\hyperOlverF@{\NVar{a}}{\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
 
xml-id: C15.S2.E2.m2afdec
Property / Symbols used
 
Property / Symbols used: Q11644 / rank
 
Normal rank
Property / Symbols used: Q11644 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11644 / qualifier
 
xml-id: C15.S1.XMD3.m1gdec
Property / Symbols used
 
Property / Symbols used: Q11645 / rank
 
Normal rank
Property / Symbols used: Q11645 / qualifier
 
test:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11645 / qualifier
 
xml-id: C15.S1.XMD4.m1fdec
Property / Symbols used
 
Property / Symbols used: Q11646 / rank
 
Normal rank
Property / Symbols used: Q11646 / qualifier
 
test:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q11646 / qualifier
 
xml-id: C15.S1.XMD5.m1ddec

Latest revision as of 13:12, 2 January 2020

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Language Label Description Also known as
English
DLMF:15.9.E22
No description defined

    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}}
    0 references
    a , b ≠ - 1 2 , - 3 2 , - 5 2 , … formulae-sequence 𝑎 𝑏 1 2 3 2 5 2 … {\displaystyle{\displaystyle a,b\neq-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},% \ldots}}
    0 references
    0 < | ph ⁡ z | < π 0 ph 𝑧 𝜋 {\displaystyle{\displaystyle 0<|\operatorname{ph}z|<\pi}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    Q11481
    test
    P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1afdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1afdec
    0 references
    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    test
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2afdec
    0 references
    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1gdec
    0 references
    Q11645
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1fdec
    0 references
    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1ddec
    0 references
     
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