DLMF:30.11.E12 (Q8907)

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DLMF:30.11.E12
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    Statements

    Defining formula
    A n - m ⁢ ( γ 2 ) ⁢ S n m ⁢ ( 1 ) ⁡ ( z , γ ) = 1 2 ⁢ i m + n ⁢ γ m ⁢ ( n - m ) ! ( n + m ) ! ⁢ z m ⁢ ( 1 - z - 2 ) 1 2 ⁢ m ⁢ ∫ - 1 1 e - i ⁢ γ ⁢ z ⁢ t ⁢ ( 1 - t 2 ) 1 2 ⁢ m ⁢ 𝖯𝗌 n m ⁡ ( t , γ 2 ) ⁢ d t . superscript subscript 𝐴 𝑛 𝑚 superscript 𝛾 2 radial-spheroidal-wave-S 𝑚 1 𝑛 𝑧 𝛾 1 2 imaginary-unit 𝑚 𝑛 superscript 𝛾 𝑚 𝑛 𝑚 𝑛 𝑚 superscript 𝑧 𝑚 superscript 1 superscript 𝑧 2 1 2 𝑚 superscript subscript 1 1 superscript 𝑒 imaginary-unit 𝛾 𝑧 𝑡 superscript 1 superscript 𝑡 2 1 2 𝑚 spheroidal-wave-on-cut-Ps 𝑚 𝑛 𝑡 superscript 𝛾 2 𝑡 {\displaystyle{\displaystyle A_{n}^{-m}(\gamma^{2})S^{m(1)}_{n}\left(z,\gamma% \right)=\frac{1}{2}{\mathrm{i}^{m+n}}\gamma^{m}\frac{(n-m)!}{(n+m)!}z^{m}(1-z^% {-2})^{\frac{1}{2}m}\*\int_{-1}^{1}e^{-\mathrm{i}\gamma zt}(1-t^{2})^{\frac{1}% {2}m}\mathsf{Ps}^{m}_{n}\left(t,\gamma^{2}\right)\mathrm{d}t.}}
    0 references
    DLMF id
    DLMF:30.11.E12
    0 references
    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    radial spheroidal wave function
    Defining formula
    S n m ⁢ ( j ) ⁡ ( z , γ ) radial-spheroidal-wave-S 𝑚 𝑗 𝑛 𝑧 𝛾 {\displaystyle{\displaystyle S^{\NVar{m}(\NVar{j})}_{\NVar{n}}\left(\NVar{z},% \NVar{\gamma}\right)}}
    xml-id
    C30.S11.E3.m2ajdec
    0 references
    Q12418
    Defining formula
    𝖯𝗌 n m ⁡ ( x , γ 2 ) spheroidal-wave-on-cut-Ps 𝑚 𝑛 𝑥 superscript 𝛾 2 {\displaystyle{\displaystyle\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},% \NVar{\gamma^{2}}\right)}}
    xml-id
    C30.S4.SS1.m1abdec
    0 references
    Q12401
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C30.S1.XMD1.m1hdec
    0 references
    Q12408
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C30.S1.XMD4.m1idec
    0 references
     
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