DLMF:15.4.E29 (Q5013): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11646 / rank
 
Normal rank
Property / Symbols used: Q11646 / qualifier
 
Defining formula:

b 𝑏 {\displaystyle{\displaystyle b}}

b
Property / Symbols used: Q11646 / qualifier
 
xml-id: C15.S1.XMD5.m1jdec

Latest revision as of 13:51, 2 January 2020

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DLMF:15.4.E29
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    Statements

    Defining formula
    F ⁡ ( a , b ; 1 2 ⁢ a + 1 2 ⁢ b + 1 ; 1 2 ) = 2 ⁢ π a - b ⁢ Γ ⁡ ( 1 2 ⁢ a + 1 2 ⁢ b + 1 ) ⁢ ( 1 Γ ⁡ ( 1 2 ⁢ a ) ⁢ Γ ⁡ ( 1 2 ⁢ b + 1 2 ) - 1 Γ ⁡ ( 1 2 ⁢ a + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ b ) ) . Gauss-hypergeometric-F 𝑎 𝑏 1 2 𝑎 1 2 𝑏 1 1 2 2 𝜋 𝑎 𝑏 Euler-Gamma 1 2 𝑎 1 2 𝑏 1 1 Euler-Gamma 1 2 𝑎 Euler-Gamma 1 2 𝑏 1 2 1 Euler-Gamma 1 2 𝑎 1 2 Euler-Gamma 1 2 𝑏 {\displaystyle{\displaystyle F\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+1;\tfrac{1% }{2}\right)=\frac{2\sqrt{\pi}}{a-b}\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b+1% \right)\*\left(\frac{1}{\Gamma\left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{% 2}b+\tfrac{1}{2}\right)}-\frac{1}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right% )\Gamma\left(\tfrac{1}{2}b\right)}\right).}}
    0 references
    DLMF id
    DLMF:15.4.E29
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2agdec
    0 references
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    Defining formula
    F ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
    xml-id
    C15.S2.E1.m2aaadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1vdec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1jdec
    0 references
     
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