DLMF:14.18.E4 (Q4905): 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
 
test:

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

\cosh@@{\NVar{z}}
Property / Symbols used: hyperbolic cosine function / qualifier
 
xml-id: C4.S28.E2.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.m2adec
Property / Symbols used
 
Property / Symbols used: Q11447 / rank
 
Normal rank
Property / Symbols used: Q11447 / qualifier
 
test:

P ν ⁡ ( z ) = P ν 0 ⁡ ( z ) shorthand-Legendre-P-first-kind 𝜈 𝑧 Legendre-P-first-kind 0 𝜈 𝑧 {\displaystyle{\displaystyle P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}% \left(z\right)}}

\assLegendreP[]{\NVar{\nu}}@{\NVar{z}}=\assLegendreP[0]{\nu}@{z}
Property / Symbols used: Q11447 / qualifier
 
xml-id: C14.S2.SS2.p2.m3adec
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.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11590 / rank
 
Normal rank
Property / Symbols used: Q11590 / qualifier
 
test:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11590 / qualifier
 
xml-id: C14.S1.XMD8.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11624 / rank
 
Normal rank
Property / Symbols used: Q11624 / qualifier
 
test:

ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}

\phi
Property / Symbols used: Q11624 / qualifier
 
xml-id: C14.S18.XMD2.m1cdec

Latest revision as of 12:27, 2 January 2020

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DLMF:14.18.E4
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    Statements

    test
    P ν ⁡ ( cosh ⁡ ξ 1 ⁢ cosh ⁡ ξ 2 - sinh ⁡ ξ 1 ⁢ sinh ⁡ ξ 2 ⁢ cos ⁡ ϕ ) = P ν ⁡ ( cosh ⁡ ξ 1 ) ⁢ P ν ⁡ ( cosh ⁡ ξ 2 ) + 2 ⁢ ∑ m = 1 ∞ ( - 1 ) m ⁢ P ν - m ⁡ ( cosh ⁡ ξ 1 ) ⁢ P ν m ⁡ ( cosh ⁡ ξ 2 ) ⁢ cos ⁡ ( m ⁢ ϕ ) , shorthand-Legendre-P-first-kind 𝜈 subscript 𝜉 1 subscript 𝜉 2 subscript 𝜉 1 subscript 𝜉 2 italic-ϕ shorthand-Legendre-P-first-kind 𝜈 subscript 𝜉 1 shorthand-Legendre-P-first-kind 𝜈 subscript 𝜉 2 2 superscript subscript 𝑚 1 superscript 1 𝑚 Legendre-P-first-kind 𝑚 𝜈 subscript 𝜉 1 Legendre-P-first-kind 𝑚 𝜈 subscript 𝜉 2 𝑚 italic-ϕ {\displaystyle{\displaystyle P_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1% }\sinh\xi_{2}\cos\phi\right)=P_{\nu}\left(\cosh\xi_{1}\right)P_{\nu}\left(% \cosh\xi_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}% \right)P^{m}_{\nu}\left(\cosh\xi_{2}\right)\cos\left(m\phi\right),}}
    0 references
    DLMF id
    DLMF:14.18.E4
    0 references
    Symbols used
    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.m1adec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    0 references
    hyperbolic cosine function
    test
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2adec
    0 references
    hyperbolic sine function
    test
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
    0 references
    Q11447
    test
    P ν ⁡ ( z ) = P ν 0 ⁡ ( z ) shorthand-Legendre-P-first-kind 𝜈 𝑧 Legendre-P-first-kind 0 𝜈 𝑧 {\displaystyle{\displaystyle P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}% \left(z\right)}}
    xml-id
    C14.S2.SS2.p2.m3adec
    0 references
    Q11593
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S1.XMD5.m1cdec
    0 references
    Q11590
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1bdec
    0 references
    Q11624
    test
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C14.S18.XMD2.m1cdec
    0 references
     
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