DLMF:15.4.E33 (Q5017): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2adec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

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

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2adec
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.m1zdec

Latest revision as of 13:51, 2 January 2020

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DLMF:15.4.E33
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    Statements

    Defining formula
    F ⁡ ( 3 ⁢ a , 1 3 + a ; 2 3 + 2 ⁢ a ; e i ⁢ π / 3 ) = π ⁢ e i ⁢ π ⁢ a / 2 ⁢ ( 16 27 ) ( 3 ⁢ a + 1 ) / 6 ⁢ Γ ⁡ ( 5 6 + a ) Γ ⁡ ( 2 3 + a ) ⁢ Γ ⁡ ( 2 3 ) , Gauss-hypergeometric-F 3 𝑎 1 3 𝑎 2 3 2 𝑎 superscript 𝑒 imaginary-unit 𝜋 3 𝜋 superscript 𝑒 imaginary-unit 𝜋 𝑎 2 superscript 16 27 3 𝑎 1 6 Euler-Gamma 5 6 𝑎 Euler-Gamma 2 3 𝑎 Euler-Gamma 2 3 {\displaystyle{\displaystyle F\left(3a,\tfrac{1}{3}+a;\tfrac{2}{3}+2a;e^{% \ifrac{\mathrm{i}\pi}{3}}\right)=\sqrt{\pi}e^{\ifrac{\mathrm{i}\pi a}{2}}\left% (\frac{16}{27}\right)^{(3a+1)/6}\frac{\Gamma\left(\frac{5}{6}+a\right)}{\Gamma% \left(\frac{2}{3}+a\right)\Gamma\left(\frac{2}{3}\right)},}}
    0 references
    DLMF id
    DLMF:15.4.E33
    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.m2akdec
    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.m2aaedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1zdec
    0 references
     
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