DLMF:14.19.E7 (Q4920): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11481 / rank
 
Normal rank
Property / Symbols used: Q11481 / qualifier
 
test:

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.m1abdec
Property / Symbols used
 
Property / Symbols used: Q11592 / rank
 
Normal rank
Property / Symbols used: Q11592 / qualifier
 
test:

𝑸 ν μ ⁡ ( z ) associated-Legendre-black-Q 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(% \NVar{z}\right)}}

\assLegendreOlverQ[\NVar{\mu}]{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Q11592 / qualifier
 
xml-id: C14.S21.SS1.p1.m3acdec
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
 
test:

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.m2addec
Property / Symbols used
 
Property / Symbols used: hyperbolic cosine function / rank
 
Normal rank
Property / Symbols used: hyperbolic cosine function / qualifier
 
test:

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

\cosh@@{\NVar{z}}
Property / Symbols used: hyperbolic cosine function / qualifier
 
xml-id: C4.S28.E2.m2afdec
Property / Symbols used
 
Property / Symbols used: hyperbolic cotangent function / rank
 
Normal rank
Property / Symbols used: hyperbolic cotangent function / qualifier
 
test:

coth ⁡ z hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\coth\NVar{z}}}

\coth@@{\NVar{z}}
Property / Symbols used: hyperbolic cotangent function / qualifier
 
xml-id: C4.S28.E7.m2adec
Property / Symbols used
 
Property / Symbols used: hyperbolic sine function / rank
 
Normal rank
Property / Symbols used: hyperbolic sine function / qualifier
 
test:

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

\sinh@@{\NVar{z}}
Property / Symbols used: hyperbolic sine function / qualifier
 
xml-id: C4.S28.E1.m2addec
Property / Symbols used
 
Property / Symbols used: Q11593 / rank
 
Normal rank
Property / Symbols used: Q11593 / qualifier
 
test:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11593 / qualifier
 
xml-id: C14.S1.XMD5.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11602 / rank
 
Normal rank
Property / Symbols used: Q11602 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11602 / qualifier
 
xml-id: C14.S1.XMD6.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11629 / rank
 
Normal rank
Property / Symbols used: Q11629 / qualifier
 
test:

ξ > 0 𝜉 0 {\displaystyle{\displaystyle\xi>0}}

\xi>0
Property / Symbols used: Q11629 / qualifier
 
xml-id: C14.S19.XMD10.m1dec

Latest revision as of 12:30, 2 January 2020

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DLMF:14.19.E7
No description defined

    Statements

    test
    P n - 1 2 m ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( n + m + 1 2 ) Γ ⁡ ( n - m + 1 2 ) ⁢ ( 2 π ⁢ sinh ⁡ ξ ) 1 / 2 ⁢ 𝑸 m - 1 2 n ⁡ ( coth ⁡ ξ ) , Legendre-P-first-kind 𝑚 𝑛 1 2 𝜉 Euler-Gamma 𝑛 𝑚 1 2 Euler-Gamma 𝑛 𝑚 1 2 superscript 2 𝜋 𝜉 1 2 associated-Legendre-black-Q 𝑛 𝑚 1 2 hyperbolic-cotangent 𝜉 {\displaystyle{\displaystyle P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{% \Gamma\left(n+m+\tfrac{1}{2}\right)}{\Gamma\left(n-m+\tfrac{1}{2}\right)}\*% \left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\boldsymbol{Q}^{n}_{m-\frac{1}{2}}% \left(\coth\xi\right),}}
    0 references
    DLMF id
    DLMF:14.19.E7
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
    Q11481
    test
    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.m1abdec
    0 references
    Q11592
    test
    𝑸 ν μ ⁡ ( z ) associated-Legendre-black-Q 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(% \NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m3acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
    0 references
    hyperbolic cosine function
    test
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2afdec
    0 references
    hyperbolic cotangent function
    test
    coth ⁡ z hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\coth\NVar{z}}}
    xml-id
    C4.S28.E7.m2adec
    0 references
    hyperbolic sine function
    test
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2addec
    0 references
    Q11593
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S1.XMD5.m1bdec
    0 references
    Q11602
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C14.S1.XMD6.m1cdec
    0 references
    Q11629
    test
    ξ > 0 𝜉 0 {\displaystyle{\displaystyle\xi>0}}
    xml-id
    C14.S19.XMD10.m1dec
    0 references
     
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