DLMF:15.9.E16 (Q5106): Difference between revisions

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Property / constraint
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| 1 - z | < 1 1 𝑧 1 {\displaystyle{\displaystyle|1-z|<1}}

|1-z|<1
Property / constraint: | 1 - z | < 1 1 𝑧 1 {\displaystyle{\displaystyle|1-z|<1}} / rank
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Normal rank
Property / Symbols used
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Property / Symbols used: gamma function / rank
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Normal rank
Property / Symbols used: gamma function / qualifier
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test:

Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
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xml-id: C5.S2.E1.m2abdec
Property / Symbols used
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Property / Symbols used: Q11481 / rank
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Normal rank
Property / Symbols used: Q11481 / qualifier
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test:

P Ξ½ ΞΌ ⁑ ( z ) Legendre-P-first-kind πœ‡ 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
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xml-id: C14.S21.SS1.p1.m1adec
Property / Symbols used
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Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
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Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
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test:

Ο€ {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
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xml-id: C3.S12.E1.m2aadec
Property / Symbols used
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Property / Symbols used: Q10817 / rank
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Normal rank
Property / Symbols used: Q10817 / qualifier
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test:

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

\phase
Property / Symbols used: Q10817 / qualifier
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xml-id: C1.S9.E7.m1adec
Property / Symbols used
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Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / rank
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Normal rank
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
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test:

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)}}

\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
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xml-id: C15.S2.E2.m2adec
Property / Symbols used
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Property / Symbols used: Q11644 / rank
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Normal rank
Property / Symbols used: Q11644 / qualifier
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test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11644 / qualifier
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xml-id: C15.S1.XMD3.m1adec
Property / Symbols used
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Property / Symbols used: Q11645 / rank
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Normal rank
Property / Symbols used: Q11645 / qualifier
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test:

a π‘Ž {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11645 / qualifier
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xml-id: C15.S1.XMD4.m1dec
Property / Symbols used
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Property / Symbols used: Q11646 / rank
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Normal rank
Property / Symbols used: Q11646 / qualifier
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test:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q11646 / qualifier
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xml-id: C15.S1.XMD5.m1dec

Latest revision as of 13:10, 2 January 2020

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DLMF:15.9.E16
No description defined

    Statements

    test
    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),}}
    0 references
    DLMF id
    DLMF:15.9.E16
    0 references
    constraint
    | ph ⁑ ( 1 - z ) | < Ο€ phase 1 𝑧 πœ‹ {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
    0 references
    | ph ⁑ ( 1 - z ) | < Ο€ ph 1 𝑧 πœ‹ {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
    0 references
    | 1 - z | < 1 1 𝑧 1 {\displaystyle{\displaystyle|1-z|<1}}
    0 references
    Symbols used
    gamma function
    test
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    Q11481
    test
    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
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
    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 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
    0 references
    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1adec
    0 references
    Q11645
    test
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1dec
    0 references
    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1dec
    0 references
     
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