DLMF:15.8.E26 (Q5083): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / constraint
 

| ph ⁡ ( 1 - z ) | < π phase 1 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}

|\phase@{1-z}|<\pi
Property / constraint: | ph ⁡ ( 1 - z ) | < π phase 1 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}} / rank
 
Normal rank

Revision as of 17:56, 30 December 2019

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DLMF:15.8.E26
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    Statements

    Defining formula
    F ⁡ ( a , 1 - a c ; z ) = ( 1 - z ) c - 1 ⁢ Γ ⁡ ( c ) ⁢ Γ ⁡ ( 1 2 ) Γ ⁡ ( 1 2 ⁢ ( c - a + 1 ) ) ⁢ Γ ⁡ ( 1 2 ⁢ c + 1 2 ⁢ a ) ⁢ F ⁡ ( 1 2 ⁢ c - 1 2 ⁢ a , 1 2 ⁢ c + 1 2 ⁢ a - 1 2 1 2 ; ( 1 - 2 ⁢ z ) 2 ) + ( 1 - 2 ⁢ z ) ⁢ ( 1 - z ) c - 1 ⁢ Γ ⁡ ( c ) ⁢ Γ ⁡ ( - 1 2 ) Γ ⁡ ( 1 2 ⁢ c - 1 2 ⁢ a ) ⁢ Γ ⁡ ( 1 2 ⁢ ( c + a - 1 ) ) ⁢ F ⁡ ( 1 2 ⁢ c - 1 2 ⁢ a + 1 2 , 1 2 ⁢ c + 1 2 ⁢ a 3 2 ; ( 1 - 2 ⁢ z ) 2 ) , Gauss-hypergeometric-F 𝑎 1 𝑎 𝑐 𝑧 superscript 1 𝑧 𝑐 1 Euler-Gamma 𝑐 Euler-Gamma 1 2 Euler-Gamma 1 2 𝑐 𝑎 1 Euler-Gamma 1 2 𝑐 1 2 𝑎 Gauss-hypergeometric-F 1 2 𝑐 1 2 𝑎 1 2 𝑐 1 2 𝑎 1 2 1 2 superscript 1 2 𝑧 2 1 2 𝑧 superscript 1 𝑧 𝑐 1 Euler-Gamma 𝑐 Euler-Gamma 1 2 Euler-Gamma 1 2 𝑐 1 2 𝑎 Euler-Gamma 1 2 𝑐 𝑎 1 Gauss-hypergeometric-F 1 2 𝑐 1 2 𝑎 1 2 1 2 𝑐 1 2 𝑎 3 2 superscript 1 2 𝑧 2 {\displaystyle{\displaystyle F\left({a,1-a\atop c};z\right)=(1-z)^{c-1}\frac{% \Gamma\left(c\right)\Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}(% c-a+1)\right)\Gamma\left(\tfrac{1}{2}c+\tfrac{1}{2}a\right)}F\left({\tfrac{1}{% 2}c-\tfrac{1}{2}a,\tfrac{1}{2}c+\tfrac{1}{2}a-\tfrac{1}{2}\atop\tfrac{1}{2}};(% 1-2z)^{2}\right)+(1-2z)(1-z)^{c-1}\frac{\Gamma\left(c\right)\Gamma\left(-% \tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}c-\tfrac{1}{2}a\right)\Gamma\left% (\tfrac{1}{2}(c+a-1)\right)}F\left({\tfrac{1}{2}c-\tfrac{1}{2}a+\tfrac{1}{2},% \tfrac{1}{2}c+\tfrac{1}{2}a\atop\tfrac{3}{2}};(1-2z)^{2}\right),}}
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    DLMF id
    DLMF:15.8.E26
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    constraint
    | ph ⁡ z | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\pi}}
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    | ph ⁡ ( 1 - z ) | < π phase 1 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(1-z\right)|<\pi}}
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