DLMF:30.11.E6 (Q8901): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q12408 / rank
 
Normal rank
Property / Symbols used: Q12408 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q12408 / qualifier
 
xml-id: C30.S1.XMD4.m1cdec
Property / Symbols used
 
Property / Symbols used: Q12409 / rank
 
Normal rank
Property / Symbols used: Q12409 / qualifier
 
Defining formula:

n ≥ m 𝑛 𝑚 {\displaystyle{\displaystyle n\geq m}}

n\geq m
Property / Symbols used: Q12409 / qualifier
 
xml-id: C30.S1.XMD5.m1gdec
Property / Symbols used
 
Property / Symbols used: spherical function (locally) / rank
 
Normal rank
Property / Symbols used: spherical function (locally) / qualifier
 
Defining formula:

ψ k ( j ) ⁢ ( z ) superscript subscript 𝜓 𝑘 𝑗 𝑧 {\displaystyle{\displaystyle\psi_{k}^{(j)}(z)}}

\psi_{k}^{(j)}(z)
Property / Symbols used: spherical function (locally) / qualifier
 
xml-id: C30.S11.XMD3.m1bdec
Property / Symbols used
 
Property / Symbols used: Q12404 / rank
 
Normal rank
Property / Symbols used: Q12404 / qualifier
 
Defining formula:

γ 2 superscript 𝛾 2 {\displaystyle{\displaystyle\gamma^{2}}}

\gamma^{2}
Property / Symbols used: Q12404 / qualifier
 
xml-id: C30.S2.XMD3.m1cdec

Latest revision as of 01:48, 2 January 2020

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DLMF:30.11.E6
No description defined

    Statements

    Defining formula
    S n m ⁢ ( j ) ⁡ ( z , γ ) = { ψ n ( j ) ⁢ ( γ ⁢ z ) + O ⁡ ( z - 2 ⁢ e | ℑ ⁡ z | ) , j = 1 , 2 , ψ n ( j ) ⁢ ( γ ⁢ z ) ⁢ ( 1 + O ⁡ ( z - 1 ) ) , j = 3 , 4 . radial-spheroidal-wave-S 𝑚 𝑗 𝑛 𝑧 𝛾 cases superscript subscript 𝜓 𝑛 𝑗 𝛾 𝑧 Big-O superscript 𝑧 2 superscript 𝑒 𝑧 𝑗 1 2 superscript subscript 𝜓 𝑛 𝑗 𝛾 𝑧 1 Big-O superscript 𝑧 1 𝑗 3 4 {\displaystyle{\displaystyle S^{m(j)}_{n}\left(z,\gamma\right)=\begin{cases}% \psi_{n}^{(j)}(\gamma z)+O\left(z^{-2}e^{|\Im z|}\right),&j=1,2,\\ \psi_{n}^{(j)}(\gamma z)\left(1+O\left(z^{-1}\right)\right),&j=3,4.\end{cases}}}
    0 references
    DLMF id
    DLMF:30.11.E6
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10901
    Defining formula
    ℑ ⁡ absent {\displaystyle{\displaystyle\Im}}
    xml-id
    C1.S9.E2.m2adec
    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.m2afdec
    0 references
    Q12401
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C30.S1.XMD1.m1cdec
    0 references
    Q12408
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C30.S1.XMD4.m1cdec
    0 references
    Q12409
    Defining formula
    n ≥ m 𝑛 𝑚 {\displaystyle{\displaystyle n\geq m}}
    xml-id
    C30.S1.XMD5.m1gdec
    0 references
    spherical function (locally)
    Defining formula
    ψ k ( j ) ⁢ ( z ) superscript subscript 𝜓 𝑘 𝑗 𝑧 {\displaystyle{\displaystyle\psi_{k}^{(j)}(z)}}
    xml-id
    C30.S11.XMD3.m1bdec
    0 references
    Q12404
    Defining formula
    γ 2 superscript 𝛾 2 {\displaystyle{\displaystyle\gamma^{2}}}
    xml-id
    C30.S2.XMD3.m1cdec
    0 references
     
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