DLMF:14.19.E8 (Q4921): Difference between revisions

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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.m2agdec
Property / Symbols used
 
Property / Symbols used: hyperbolic cotangent function / rank
 
Normal rank
Property / Symbols used: hyperbolic cotangent function / qualifier
 
Defining formula:

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.m2aadec
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.m2aedec
Property / Symbols used
 
Property / Symbols used: Q11593 / rank
 
Normal rank
Property / Symbols used: Q11593 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11593 / qualifier
 
xml-id: C14.S1.XMD5.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11602 / rank
 
Normal rank
Property / Symbols used: Q11602 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11602 / qualifier
 
xml-id: C14.S1.XMD6.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11629 / rank
 
Normal rank
Property / Symbols used: Q11629 / qualifier
 
Defining formula:

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

\xi>0
Property / Symbols used: Q11629 / qualifier
 
xml-id: C14.S19.XMD10.m1adec

Latest revision as of 13:31, 2 January 2020

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DLMF:14.19.E8
No description defined

    Statements

    Defining formula
    𝑸 n - 1 2 m ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( m - n + 1 2 ) Γ ⁡ ( m + n + 1 2 ) ⁢ ( π 2 ⁢ sinh ⁡ ξ ) 1 / 2 ⁢ P m - 1 2 n ⁡ ( coth ⁡ ξ ) . associated-Legendre-black-Q 𝑚 𝑛 1 2 𝜉 Euler-Gamma 𝑚 𝑛 1 2 Euler-Gamma 𝑚 𝑛 1 2 superscript 𝜋 2 𝜉 1 2 Legendre-P-first-kind 𝑛 𝑚 1 2 hyperbolic-cotangent 𝜉 {\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi% \right)=\frac{\Gamma\left(m-n+\tfrac{1}{2}\right)}{\Gamma\left(m+n+\tfrac{1}{2% }\right)}\*\left(\frac{\pi}{2\sinh\xi}\right)^{1/2}P^{n}_{m-\frac{1}{2}}\left(% \coth\xi\right).}}
    0 references
    DLMF id
    DLMF:14.19.E8
    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.m2aedec
    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.m1acdec
    0 references
    Q11592
    Defining formula
    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.m3addec
    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
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2agdec
    0 references
    hyperbolic cotangent function
    Defining formula
    coth ⁡ z hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\coth\NVar{z}}}
    xml-id
    C4.S28.E7.m2aadec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2aedec
    0 references
    Q11593
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S1.XMD5.m1cdec
    0 references
    Q11602
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C14.S1.XMD6.m1ddec
    0 references
    Q11629
    Defining formula
    ξ > 0 𝜉 0 {\displaystyle{\displaystyle\xi>0}}
    xml-id
    C14.S19.XMD10.m1adec
    0 references
     
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