DLMF:15.8.E24 (Q5081): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
 
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Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
Defining formula:

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

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.m1andec
Property / Symbols used
 
Property / Symbols used: Q11644 / rank
 
Normal rank
Property / Symbols used: Q11644 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11644 / qualifier
 
xml-id: C15.S1.XMD3.m1wdec
Property / Symbols used
 
Property / Symbols used: Q11645 / rank
 
Normal rank
Property / Symbols used: Q11645 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11645 / qualifier
 
xml-id: C15.S1.XMD4.m1vdec
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.m1sdec

Latest revision as of 14:06, 2 January 2020

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DLMF:15.8.E24
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    Statements

    F ( a , b a - b + 1 ; z ) = ( 1 - z ) - a Γ ( a - b + 1 ) Γ ( 1 2 ) Γ ( 1 2 a + 1 2 ) Γ ( 1 2 a - b + 1 ) F ( 1 2 a , 1 2 a - b + 1 2 1 2 ; ( z + 1 z - 1 ) 2 ) + ( 1 + z ) ( 1 - z ) - a - 1 Γ ( a - b + 1 ) Γ ( - 1 2 ) Γ ( 1 2 a ) Γ ( 1 2 a - b + 1 2 ) F ( 1 2 a + 1 2 , 1 2 a - b + 1 3 2 ; ( z + 1 z - 1 ) 2 ) , Gauss-hypergeometric-F 𝑎 𝑏 𝑎 𝑏 1 𝑧 superscript 1 𝑧 𝑎 Euler-Gamma 𝑎 𝑏 1 Euler-Gamma 1 2 Euler-Gamma 1 2 𝑎 1 2 Euler-Gamma 1 2 𝑎 𝑏 1 Gauss-hypergeometric-F 1 2 𝑎 1 2 𝑎 𝑏 1 2 1 2 superscript 𝑧 1 𝑧 1 2 1 𝑧 superscript 1 𝑧 𝑎 1 Euler-Gamma 𝑎 𝑏 1 Euler-Gamma 1 2 Euler-Gamma 1 2 𝑎 Euler-Gamma 1 2 𝑎 𝑏 1 2 Gauss-hypergeometric-F 1 2 𝑎 1 2 1 2 𝑎 𝑏 1 3 2 superscript 𝑧 1 𝑧 1 2 {\displaystyle{\displaystyle F\left({a,b\atop a-b+1};z\right)=(1-z)^{-a}\frac{% \Gamma\left(a-b+1\right)\Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}% {2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}a-b+1\right)}F\left({\tfrac{1}% {2}a,\tfrac{1}{2}a-b+\tfrac{1}{2}\atop\tfrac{1}{2}};\left(\frac{z+1}{z-1}% \right)^{2}\right)+(1+z)(1-z)^{-a-1}\frac{\Gamma\left(a-b+1\right)\Gamma\left(% -\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}% a-b+\tfrac{1}{2}\right)}F\left({\tfrac{1}{2}a+\tfrac{1}{2},\tfrac{1}{2}a-b+1% \atop\tfrac{3}{2}};\left(\frac{z+1}{z-1}\right)^{2}\right),}}
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    DLMF:15.8.E24
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    | ph ( - z ) | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\pi}}
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    Γ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    C5.S2.E1.m2ahdec
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    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)}}
    C15.S2.E1.m2amdec
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    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2andec
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    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    C1.S9.E7.m1andec
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    z 𝑧 {\displaystyle{\displaystyle z}}
    C15.S1.XMD3.m1wdec
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    a 𝑎 {\displaystyle{\displaystyle a}}
    C15.S1.XMD4.m1vdec
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    b 𝑏 {\displaystyle{\displaystyle b}}
    C15.S1.XMD5.m1sdec
    0 references