DLMF:14.18.E5 (Q4906)

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DLMF:14.18.E5
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    Statements

    Defining formula
    Q ν ⁡ ( cosh ⁡ ξ 1 ⁢ cosh ⁡ ξ 2 - sinh ⁡ ξ 1 ⁢ sinh ⁡ ξ 2 ⁢ cos ⁡ ϕ ) = P ν ⁡ ( cosh ⁡ ξ 1 ) ⁢ Q ν ⁡ ( cosh ⁡ ξ 2 ) + 2 ⁢ ∑ m = 1 ∞ ( - 1 ) m ⁢ P ν - m ⁡ ( cosh ⁡ ξ 1 ) ⁢ Q ν m ⁡ ( cosh ⁡ ξ 2 ) ⁢ cos ⁡ ( m ⁢ ϕ ) . shorthand-Legendre-Q-second-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-Q-second-kind 𝜈 subscript 𝜉 2 2 superscript subscript 𝑚 1 superscript 1 𝑚 Legendre-P-first-kind 𝑚 𝜈 subscript 𝜉 1 Legendre-Q-second-kind 𝑚 𝜈 subscript 𝜉 2 𝑚 italic-ϕ {\displaystyle{\displaystyle Q_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1% }\sinh\xi_{2}\cos\phi\right)=P_{\nu}\left(\cosh\xi_{1}\right)Q_{\nu}\left(% \cosh\xi_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}% \right)Q^{m}_{\nu}\left(\cosh\xi_{2}\right)\cos\left(m\phi\right).}}
    0 references
    DLMF id
    DLMF:14.18.E5
    0 references
    Symbols used
    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.m1aadec
    0 references
    Q11470
    Defining formula
    Q ν μ ⁡ ( z ) Legendre-Q-second-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m2adec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2addec
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    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2aadec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2aadec
    0 references
    Q11447
    Defining formula
    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.m3aadec
    0 references
    Q11448
    Defining formula
    Q ν ⁡ ( z ) = Q ν 0 ⁡ ( z ) shorthand-Legendre-Q-second-kind 𝜈 𝑧 Legendre-Q-second-kind 0 𝜈 𝑧 {\displaystyle{\displaystyle Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}% \left(z\right)}}
    xml-id
    C14.S2.SS2.p2.m5adec
    0 references
    Q11593
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S1.XMD5.m1ddec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1cdec
    0 references
    Q11624
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C14.S18.XMD2.m1ddec
    0 references
    Q11625
    Defining formula
    ξ 1 subscript 𝜉 1 {\displaystyle{\displaystyle\xi_{1}}}
    xml-id
    C14.S18.XMD3.m1adec
    0 references
     
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