DLMF:14.12.E4 (Q4825): Difference between revisions

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Property / constraint
 

ℜ ⁡ ( μ - ν ) > 0 𝜇 𝜈 0 {\displaystyle{\displaystyle\Re(\mu-\nu)>0}}

\realpart@@{(\mu-\nu)}>0
Property / constraint: ℜ ⁡ ( μ - ν ) > 0 𝜇 𝜈 0 {\displaystyle{\displaystyle\Re(\mu-\nu)>0}} / rank
 
Normal rank
Property / constraint
 

ν + μ ≠ - 1 , - 2 , - 3 , … 𝜈 𝜇 1 2 3 … {\displaystyle{\displaystyle\nu+\mu\neq-1,-2,-3,\dots}}

\nu+\mu\neq-1,-2,-3,\dots
Property / constraint: ν + μ ≠ - 1 , - 2 , - 3 , … 𝜈 𝜇 1 2 3 … {\displaystyle{\displaystyle\nu+\mu\neq-1,-2,-3,\dots}} / rank
 
Normal rank
Property / constraint
 

ℜ ⁡ ( μ - ν ) > 0 𝜇 𝜈 0 {\displaystyle{\displaystyle\Re(\mu-\nu)>0}}

\Re(\mu-\nu)>0
Property / constraint: ℜ ⁡ ( μ - ν ) > 0 𝜇 𝜈 0 {\displaystyle{\displaystyle\Re(\mu-\nu)>0}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
Defining formula:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2acdec
Property / Symbols used
 
Property / Symbols used: Q11481 / rank
 
Normal rank
Property / Symbols used: Q11481 / qualifier
 
Defining formula:

P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}

\assLegendreP[\NVar{\mu}]{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Q11481 / qualifier
 
xml-id: C14.S21.SS1.p1.m1adec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1acdec
Property / Symbols used
 
Property / Symbols used: hyperbolic cosine function / rank
 
Normal rank
Property / Symbols used: hyperbolic cosine function / qualifier
 
Defining formula:

cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}

\cosh@@{\NVar{z}}
Property / Symbols used: hyperbolic cosine function / qualifier
 
xml-id: C4.S28.E2.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3acdec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1acdec
Property / Symbols used
 
Property / Symbols used: Q11589 / rank
 
Normal rank
Property / Symbols used: Q11589 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11589 / qualifier
 
xml-id: C14.S1.XMD1.m1adec
Property / Symbols used
 
Property / Symbols used: Q11591 / rank
 
Normal rank
Property / Symbols used: Q11591 / qualifier
 
Defining formula:

μ 𝜇 {\displaystyle{\displaystyle\mu}}

\mu
Property / Symbols used: Q11591 / qualifier
 
xml-id: C14.S1.XMD7.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11590 / rank
 
Normal rank
Property / Symbols used: Q11590 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11590 / qualifier
 
xml-id: C14.S1.XMD8.m1cdec

Latest revision as of 01:42, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:14.12.E4
No description defined

    Statements

    Defining formula
    P ν - μ ⁡ ( x ) = 2 1 / 2 ⁢ Γ ⁡ ( μ + 1 2 ) ⁢ ( x 2 - 1 ) μ / 2 π 1 / 2 ⁢ Γ ⁡ ( ν + μ + 1 ) ⁢ Γ ⁡ ( μ - ν ) ⁢ ∫ 0 ∞ cosh ⁡ ( ( ν + 1 2 ) ⁢ t ) ( x + cosh ⁡ t ) μ + ( 1 / 2 ) ⁢ d t , Legendre-P-first-kind 𝜇 𝜈 𝑥 superscript 2 1 2 Euler-Gamma 𝜇 1 2 superscript superscript 𝑥 2 1 𝜇 2 superscript 𝜋 1 2 Euler-Gamma 𝜈 𝜇 1 Euler-Gamma 𝜇 𝜈 superscript subscript 0 𝜈 1 2 𝑡 superscript 𝑥 𝑡 𝜇 1 2 𝑡 {\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{2^{1/2}\Gamma% \left(\mu+\frac{1}{2}\right)\left(x^{2}-1\right)^{\mu/2}}{\pi^{1/2}\Gamma\left% (\nu+\mu+1\right)\Gamma\left(\mu-\nu\right)}\*\int_{0}^{\infty}\frac{\cosh% \left(\left(\nu+\frac{1}{2}\right)t\right)}{(x+\cosh t)^{\mu+(1/2)}}\mathrm{d}% t,}}
    0 references
    DLMF id
    DLMF:14.12.E4
    0 references
    constraint
    ℜ ⁡ ( μ - ν ) > 0 𝜇 𝜈 0 {\displaystyle{\displaystyle\Re(\mu-\nu)>0}}
    0 references
    ν + μ ≠ - 1 , - 2 , - 3 , … 𝜈 𝜇 1 2 3 … {\displaystyle{\displaystyle\nu+\mu\neq-1,-2,-3,\dots}}
    0 references
    ℜ ⁡ ( μ - ν ) > 0 𝜇 𝜈 0 {\displaystyle{\displaystyle\Re(\mu-\nu)>0}}
    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.m2acdec
    0 references
    Q11481
    Defining formula
    P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1adec
    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
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1acdec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1adec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1cdec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1cdec
    0 references
     
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