DLMF:15.9.E11 (Q5101): Difference between revisions

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Property / Symbols defined
 
Property / Symbols defined: Jacobi function / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / rank
 
Normal rank
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / qualifier
 
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)}}

\hyperF@{\NVar{a}}{\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / qualifier
 
xml-id: C15.S2.E1.m2ajdec
Property / Symbols used
 
Property / Symbols used: hyperbolic sine function / rank
 
Normal rank
Property / Symbols used: hyperbolic sine function / qualifier
 
Defining formula:

sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}

\sinh@@{\NVar{z}}
Property / Symbols used: hyperbolic sine function / qualifier
 
xml-id: C4.S28.E1.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.m2abdec

Latest revision as of 14:09, 2 January 2020

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DLMF:15.9.E11
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    Statements

    Defining formula
    ϕ λ ( α , β ) ⁡ ( t ) = F ⁡ ( 1 2 ⁢ ( α + β + 1 - i ⁢ λ ) , 1 2 ⁢ ( α + β + 1 + i ⁢ λ ) α + 1 ; - sinh 2 ⁡ t ) . Jacobi-hypergeometric-phi 𝛼 𝛽 𝜆 𝑡 Gauss-hypergeometric-F 1 2 𝛼 𝛽 1 imaginary-unit 𝜆 1 2 𝛼 𝛽 1 imaginary-unit 𝜆 𝛼 1 2 𝑡 {\displaystyle{\displaystyle\phi^{(\alpha,\beta)}_{\lambda}\left(t\right)=F% \left({\tfrac{1}{2}(\alpha+\beta+1-\mathrm{i}\lambda),\tfrac{1}{2}(\alpha+% \beta+1+\mathrm{i}\lambda)\atop\alpha+1};-{\sinh^{2}}t\right).}}
    0 references
    DLMF id
    DLMF:15.9.E11
    0 references
    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ 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.m2ajdec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
     
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