DLMF:14.12.E1 (Q4822): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
(10 intermediate revisions by the same user not shown)
Property / constraint
 

ℜ ⁡ μ < 1 2 𝜇 1 2 {\displaystyle{\displaystyle\Re\mu<\tfrac{1}{2}}}

\Re\mu<\tfrac{1}{2}
Property / constraint: ℜ ⁡ μ < 1 2 𝜇 1 2 {\displaystyle{\displaystyle\Re\mu<\tfrac{1}{2}}} / 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.m2adec
Property / Symbols used
 
Property / Symbols used: Ferrers function of the first kind / rank
 
Normal rank
Property / Symbols used: Ferrers function of the first kind / qualifier
 
Defining formula:

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

\FerrersP[\NVar{\mu}]{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Ferrers function of the first kind / qualifier
 
xml-id: C14.S3.E1.m2adec
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.m2adec
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
Defining formula:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2adec
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.m1adec
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.m3adec
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.m1adec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2adec
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.m1dec
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.m1dec

Latest revision as of 01:41, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:14.12.E1
No description defined

    Statements

    Defining formula
    𝖯 ν μ ⁡ ( cos ⁡ θ ) = 2 1 / 2 ⁢ ( sin ⁡ θ ) μ π 1 / 2 ⁢ Γ ⁡ ( 1 2 - μ ) ⁢ ∫ 0 θ cos ⁡ ( ( ν + 1 2 ) ⁢ t ) ( cos ⁡ t - cos ⁡ θ ) μ + ( 1 / 2 ) ⁢ d t , Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝜃 superscript 2 1 2 superscript 𝜃 𝜇 superscript 𝜋 1 2 Euler-Gamma 1 2 𝜇 superscript subscript 0 𝜃 𝜈 1 2 𝑡 superscript 𝑡 𝜃 𝜇 1 2 𝑡 {\displaystyle{\displaystyle\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)=% \frac{2^{1/2}(\sin\theta)^{\mu}}{\pi^{1/2}\Gamma\left(\frac{1}{2}-\mu\right)}% \int_{0}^{\theta}\frac{\cos\left(\left(\nu+\frac{1}{2}\right)t\right)}{(\cos t% -\cos\theta)^{\mu+(1/2)}}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:14.12.E1
    0 references
    constraint
    ℜ ⁡ μ < 1 2 𝜇 1 2 {\displaystyle{\displaystyle\Re\mu<\tfrac{1}{2}}}
    0 references
    0 < θ < π 0 𝜃 𝜋 {\displaystyle{\displaystyle 0<\theta<\pi}}
    0 references
    ℜ ⁡ μ < 1 2 𝜇 1 2 {\displaystyle{\displaystyle\Re\mu<\tfrac{1}{2}}}
    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.m2adec
    0 references
    Ferrers function of the first kind
    Defining formula
    𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1dec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1dec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q4822&oldid=33769"