DLMF:15.9.E23 (Q5113): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11481 / rank
 
Normal rank
Property / Symbols used: Q11481 / qualifier
 
test:

P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}

\assLegendreP[\NVar{\mu}]{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Q11481 / qualifier
 
xml-id: C14.S21.SS1.p1.m1agdec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2ahdec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2acdec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
test:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2afdec
Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
test:

ph phase {\displaystyle{\displaystyle\operatorname{ph}}}

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.m1agdec
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.m2agdec
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.m1hdec
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.m1gdec
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.m1edec

Latest revision as of 13:12, 2 January 2020

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English
DLMF:15.9.E23
No description defined

    Statements

    test
    𝐅 ⁡ ( a , b 3 2 ; z ) = 2 a + b - ( 5 / 2 ) π ⁢ z ⁢ Γ ⁡ ( a - 1 2 ) ⁢ Γ ⁡ ( b - 1 2 ) ⁢ ( z - 1 ) ( - a - b + ( 3 / 2 ) ) / 2 ⁢ ( e ± π ⁢ i ⁢ ( a + b - ( 3 / 2 ) ) ⁢ P a - b - ( 1 / 2 ) - a - b + ( 3 / 2 ) ⁡ ( - z ) - P a - b - ( 1 / 2 ) - a - b + ( 3 / 2 ) ⁡ ( z ) ) , scaled-hypergeometric-bold-F 𝑎 𝑏 3 2 𝑧 superscript 2 𝑎 𝑏 5 2 𝜋 𝑧 Euler-Gamma 𝑎 1 2 Euler-Gamma 𝑏 1 2 superscript 𝑧 1 𝑎 𝑏 3 2 2 superscript 𝑒 plus-or-minus 𝜋 imaginary-unit 𝑎 𝑏 3 2 Legendre-P-first-kind 𝑎 𝑏 3 2 𝑎 𝑏 1 2 𝑧 Legendre-P-first-kind 𝑎 𝑏 3 2 𝑎 𝑏 1 2 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop\tfrac{3}{2}};z\right)=% \frac{2^{a+b-(\ifrac{5}{2})}}{\pi\sqrt{z}}\Gamma\left(a-\tfrac{1}{2}\right)% \Gamma\left(b-\tfrac{1}{2}\right)\*(z-1)^{(-a-b+(\ifrac{3}{2}))/2}\*\left(e^{% \pm\pi\mathrm{i}(a+b-(\ifrac{3}{2}))}P^{-a-b+(\ifrac{3}{2})}_{a-b-(\ifrac{1}{2% })}\left(-\sqrt{z}\right)-P^{-a-b+(\ifrac{3}{2})}_{a-b-(\ifrac{1}{2})}\left(% \sqrt{z}\right)\right),}}
    0 references
    DLMF id
    DLMF:15.9.E23
    0 references
    constraint
    0 < | ph ⁡ z | < π 0 phase 𝑧 𝜋 {\displaystyle{\displaystyle 0<|\operatorname{ph}z|<\pi}}
    0 references
    a , b ≠ 1 2 , - 1 2 , - 3 2 , … formulae-sequence 𝑎 𝑏 1 2 1 2 3 2 … {\displaystyle{\displaystyle a,b\neq\frac{1}{2},-\frac{1}{2},-\frac{3}{2},% \dots}}
    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.m2addec
    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.m1agdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2afdec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1agdec
    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.m2agdec
    0 references
    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1hdec
    0 references
    Q11645
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1gdec
    0 references
    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1edec
    0 references
     
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