DLMF:28.24.E13 (Q8410): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q12307 / rank
 
Normal rank
Property / Symbols used: Q12307 / qualifier
 
Defining formula:

Ko n ⁡ ( z , h ) modified-Mathieu-Ko 𝑛 𝑧 ℎ {\displaystyle{\displaystyle\mathrm{Ko}_{\NVar{n}}\left(\NVar{z},\NVar{h}% \right)}}

\modMathieuKo{\NVar{n}}@{\NVar{z}}{\NVar{h}}
Property / Symbols used: Q12307 / qualifier
 
xml-id: C28.S20.E20.m1aadec
Property / Symbols used
 
Property / Symbols used: Q12284 / rank
 
Normal rank
Property / Symbols used: Q12284 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q12284 / qualifier
 
xml-id: C28.S1.XMD1.m1kdec
Property / Symbols used
 
Property / Symbols used: Q12294 / rank
 
Normal rank
Property / Symbols used: Q12294 / qualifier
 
Defining formula:

h ℎ {\displaystyle{\displaystyle h}}

h
Property / Symbols used: Q12294 / qualifier
 
xml-id: C28.S1.XMD11.m1ldec
Property / Symbols used
 
Property / Symbols used: Q12272 / rank
 
Normal rank
Property / Symbols used: Q12272 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q12272 / qualifier
 
xml-id: C28.S1.XMD6.m1ldec
Property / Symbols used
 
Property / Symbols used: Q12287 / rank
 
Normal rank
Property / Symbols used: Q12287 / qualifier
 
Defining formula:

A m ⁢ ( q ) subscript 𝐴 𝑚 𝑞 {\displaystyle{\displaystyle A_{m}(q)}}

A_{m}(q)
Property / Symbols used: Q12287 / qualifier
 
xml-id: C28.S4.XMD1.m1edec

Latest revision as of 15:13, 2 January 2020

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DLMF:28.24.E13
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    Statements

    Defining formula
    Ko 2 ⁢ m + 1 ⁡ ( z , h ) = ∑ ℓ = 0 ∞ A 2 ⁢ ℓ + 1 2 ⁢ m + 1 ⁢ ( h 2 ) A 2 ⁢ s + 1 2 ⁢ m + 1 ⁢ ( h 2 ) ⁢ ( I ℓ - s ⁡ ( h ⁢ e - z ) ⁢ K ℓ + s + 1 ⁡ ( h ⁢ e z ) + I ℓ + s + 1 ⁡ ( h ⁢ e - z ) ⁢ K ℓ - s ⁡ ( h ⁢ e z ) ) . modified-Mathieu-Ko 2 𝑚 1 𝑧 ℎ superscript subscript ℓ 0 superscript subscript 𝐴 2 ℓ 1 2 𝑚 1 superscript ℎ 2 superscript subscript 𝐴 2 𝑠 1 2 𝑚 1 superscript ℎ 2 modified-Bessel-first-kind ℓ 𝑠 ℎ superscript 𝑒 𝑧 modified-Bessel-second-kind ℓ 𝑠 1 ℎ superscript 𝑒 𝑧 modified-Bessel-first-kind ℓ 𝑠 1 ℎ superscript 𝑒 𝑧 modified-Bessel-second-kind ℓ 𝑠 ℎ superscript 𝑒 𝑧 {\displaystyle{\displaystyle\mathrm{Ko}_{2m+1}\left(z,h\right)=\sum_{\ell=0}^{% \infty}\frac{A_{2\ell+1}^{2m+1}(h^{2})}{A_{2s+1}^{2m+1}(h^{2})}\left(I_{\ell-s% }\left(he^{-z}\right)K_{\ell+s+1}\left(he^{z}\right)+I_{\ell+s+1}\left(he^{-z}% \right)K_{\ell-s}\left(he^{z}\right)\right).}}
    0 references
    DLMF id
    DLMF:28.24.E13
    0 references
    Symbols used
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aldec
    0 references
    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2agdec
    0 references
    modified Bessel function of the second kind
    Defining formula
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2acdec
    0 references
    Q12307
    Defining formula
    Ko n ⁡ ( z , h ) modified-Mathieu-Ko 𝑛 𝑧 ℎ {\displaystyle{\displaystyle\mathrm{Ko}_{\NVar{n}}\left(\NVar{z},\NVar{h}% \right)}}
    xml-id
    C28.S20.E20.m1aadec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1kdec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1ldec
    0 references
    Q12272
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1ldec
    0 references
    Q12287
    Defining formula
    A m ⁢ ( q ) subscript 𝐴 𝑚 𝑞 {\displaystyle{\displaystyle A_{m}(q)}}
    xml-id
    C28.S4.XMD1.m1edec
    0 references
     
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