DLMF:15.2.E3 (Q4979): Difference between revisions

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Property / constraint
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x > 1 π‘₯ 1 {\displaystyle{\displaystyle x>1}}

x>1
Property / constraint: x > 1 π‘₯ 1 {\displaystyle{\displaystyle x>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: 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.m2adec
Property / Symbols used
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Property / Symbols used: imaginary unit / rank
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Normal rank
Property / Symbols used: imaginary unit / qualifier
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test:

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

\iunit
Property / Symbols used: imaginary unit / qualifier
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xml-id: C1.S9.E1.m2adec
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.m2aadec
Property / Symbols used
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Property / Symbols used: Q11649 / rank
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Normal rank
Property / Symbols used: Q11649 / qualifier
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test:

x π‘₯ {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11649 / qualifier
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xml-id: C15.S1.XMD1.m1dec
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.m1bdec
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.m1bdec
Property / Symbols used
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Property / Symbols used: Q11647 / rank
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Normal rank
Property / Symbols used: Q11647 / qualifier
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test:

c 𝑐 {\displaystyle{\displaystyle c}}

c
Property / Symbols used: Q11647 / qualifier
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xml-id: C15.S1.XMD6.m1bdec

Latest revision as of 12:46, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:15.2.E3
No description defined

    Statements

    test
    𝐅 ⁑ ( a , b c ; x + i ⁒ 0 ) - 𝐅 ⁑ ( a , b c ; x - i ⁒ 0 ) = 2 ⁒ Ο€ ⁒ i Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b ) ⁒ ( x - 1 ) c - a - b ⁒ 𝐅 ⁑ ( c - a , c - b c - a - b + 1 ; 1 - x ) , scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 π‘₯ imaginary-unit 0 scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 π‘₯ imaginary-unit 0 2 πœ‹ imaginary-unit Euler-Gamma π‘Ž Euler-Gamma 𝑏 superscript π‘₯ 1 𝑐 π‘Ž 𝑏 scaled-hypergeometric-bold-F 𝑐 π‘Ž 𝑐 𝑏 𝑐 π‘Ž 𝑏 1 1 π‘₯ {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop c};x+\mathrm{i}0\right)-% \mathbf{F}\left({a,b\atop c};x-\mathrm{i}0\right)=\frac{2\pi\mathrm{i}}{\Gamma% \left(a\right)\Gamma\left(b\right)}(x-1)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c% -a-b+1};1-x\right),}}
    0 references
    DLMF id
    DLMF:15.2.E3
    0 references
    constraint
    x > 1 π‘₯ 1 {\displaystyle{\displaystyle x>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
    the ratio of the circumference of a circle to its diameter
    test
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    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.m2aadec
    0 references
    Q11649
    test
    x π‘₯ {\displaystyle{\displaystyle x}}
    xml-id
    C15.S1.XMD1.m1dec
    0 references
    Q11645
    test
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1bdec
    0 references
    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1bdec
    0 references
    Q11647
    test
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1bdec
    0 references
     
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