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DLMF:30.6.E3 - MaRDI portal

DLMF:30.6.E3 (Q9554)

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DLMF:30.6.E3
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    Statements

    DLMF defining formula
    𝒲 ⁡ { 𝑃𝑠 n m ⁡ ( z , γ 2 ) , 𝑄𝑠 n m ⁡ ( z , γ 2 ) } = ( - 1 ) m ⁢ ( n + m ) ! ( 1 - z 2 ) ⁢ ( n - m ) ! ⁢ A n m ⁢ ( γ 2 ) ⁢ A n - m ⁢ ( γ 2 ) , Wronskian spheroidal-wave-Ps 𝑚 𝑛 𝑧 superscript 𝛾 2 spheroidal-wave-Qs 𝑚 𝑛 𝑧 superscript 𝛾 2 superscript 1 𝑚 𝑛 𝑚 1 superscript 𝑧 2 𝑛 𝑚 superscript subscript 𝐴 𝑛 𝑚 superscript 𝛾 2 superscript subscript 𝐴 𝑛 𝑚 superscript 𝛾 2 {\displaystyle{\displaystyle\mathscr{W}\left\{\mathit{Ps}^{m}_{n}\left(z,% \gamma^{2}\right),\mathit{Qs}^{m}_{n}\left(z,\gamma^{2}\right)\right\}=\frac{(% -1)^{m}(n+m)!}{(1-z^{2})(n-m)!}A_{n}^{m}(\gamma^{2})A_{n}^{-m}(\gamma^{2}),}}
    0 references
    Symbols used
    Wronskian
    DLMF defining formula
    𝒲 Wronskian {\displaystyle{\displaystyle\mathscr{W}}}
    xml-id
    C1.S13.Px2.p1.m3adec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    spheroidal wave function of complex argument
    DLMF defining formula
    𝑃𝑠 n m ⁡ ( z , γ 2 ) spheroidal-wave-Ps 𝑚 𝑛 𝑧 superscript 𝛾 2 {\displaystyle{\displaystyle\mathit{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},% \NVar{\gamma^{2}}\right)}}
    xml-id
    C30.S6.p1.m1abdec
    0 references
    spheroidal wave function of complex argument
    DLMF defining formula
    𝑄𝑠 n m ⁡ ( z , γ 2 ) spheroidal-wave-Qs 𝑚 𝑛 𝑧 superscript 𝛾 2 {\displaystyle{\displaystyle\mathit{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},% \NVar{\gamma^{2}}\right)}}
    xml-id
    C30.S6.p1.m2abdec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C30.S1.XMD1.m1bdec
    0 references
    nonnegative integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C30.S1.XMD4.m1bdec
    0 references
    integer degree
    DLMF defining formula
    n ≥ m 𝑛 𝑚 {\displaystyle{\displaystyle n\geq m}}
    xml-id
    C30.S1.XMD5.m1bdec
    0 references
    30.11.E4
    DLMF defining formula
    A n m ⁢ ( γ 2 ) superscript subscript 𝐴 𝑛 𝑚 superscript 𝛾 2 {\displaystyle{\displaystyle A_{n}^{m}(\gamma^{2})}}
    xml-id
    C30.S11.XMD5.m1dec
    0 references
    real parameter
    DLMF defining formula
    γ 2 superscript 𝛾 2 {\displaystyle{\displaystyle\gamma^{2}}}
    xml-id
    C30.S2.XMD3.m1adec
    0 references
     
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