DLMF:15.6.E9 (Q5047): Difference between revisions

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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.m1hdec
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.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11647 / rank
 
Normal rank
Property / Symbols used: Q11647 / qualifier
 
test:

c 𝑐 {\displaystyle{\displaystyle c}}

c
Property / Symbols used: Q11647 / qualifier
 
xml-id: C15.S1.XMD6.m1hdec

Latest revision as of 12:59, 2 January 2020

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English
DLMF:15.6.E9
No description defined

    Statements

    test
    𝐅 ⁡ ( a , b ; c ; z ) = ∫ 0 1 t d - 1 ⁢ ( 1 - t ) c - d - 1 ( 1 - z ⁢ t ) a + b - λ ⁢ 𝐅 ⁡ ( λ - a , λ - b d ; z ⁢ t ) ⁢ 𝐅 ⁡ ( a + b - λ , λ - d c - d ; ( 1 - t ) ⁢ z 1 - z ⁢ t ) ⁢ d t , scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 superscript subscript 0 1 superscript 𝑡 𝑑 1 superscript 1 𝑡 𝑐 𝑑 1 superscript 1 𝑧 𝑡 𝑎 𝑏 𝜆 scaled-hypergeometric-bold-F 𝜆 𝑎 𝜆 𝑏 𝑑 𝑧 𝑡 scaled-hypergeometric-bold-F 𝑎 𝑏 𝜆 𝜆 𝑑 𝑐 𝑑 1 𝑡 𝑧 1 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle\mathbf{F}\left(a,b;c;z\right)=\int_{0}^{1}\frac{t% ^{d-1}(1-t)^{c-d-1}}{(1-zt)^{a+b-\lambda}}\mathbf{F}\left({\lambda-a,\lambda-b% \atop d};zt\right)\mathbf{F}\left({a+b-\lambda,\lambda-d\atop c-d};\frac{(1-t)% z}{1-zt}\right)\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:15.6.E9
    0 references
    constraint
    | ph ⁡ ( 1 - z ) | < π phase 1 𝑧 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
    0 references
    λ ∈ ℂ 𝜆 {\displaystyle{\displaystyle\lambda\in\mathbb{C}}}
    0 references
    ℜ ⁡ c > ℜ ⁡ d > 0 𝑐 𝑑 0 {\displaystyle{\displaystyle\Re c>\Re d>0}}
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
    0 references
    Q10923
    test
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2adec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
    0 references
    Q10784
    test
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ahdec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1ahdec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aedec
    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.m2ahdec
    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.m1hdec
    0 references
    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1hdec
    0 references
    Q11647
    test
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1hdec
    0 references
     
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