DLMF:28.24.E9 (Q8406): Difference between revisions

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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.m1hdec
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.m1hdec
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.m1cdec

Latest revision as of 15:12, 2 January 2020

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DLMF:28.24.E9
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    Statements

    Defining formula
    Io 2 ⁢ m + 1 ⁡ ( z , h ) = ( - 1 ) s ⁢ ∑ ℓ = 0 ∞ ( - 1 ) ℓ ⁢ A 2 ⁢ ℓ + 1 2 ⁢ m + 1 ⁢ ( h 2 ) A 2 ⁢ s + 1 2 ⁢ m + 1 ⁢ ( h 2 ) ⁢ ( I ℓ - s ⁡ ( h ⁢ e - z ) ⁢ I ℓ + s + 1 ⁡ ( h ⁢ e z ) - I ℓ + s + 1 ⁡ ( h ⁢ e - z ) ⁢ I ℓ - s ⁡ ( h ⁢ e z ) ) , modified-Mathieu-Io 2 𝑚 1 𝑧 ℎ superscript 1 𝑠 superscript subscript ℓ 0 superscript 1 ℓ 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-first-kind ℓ 𝑠 1 ℎ superscript 𝑒 𝑧 modified-Bessel-first-kind ℓ 𝑠 1 ℎ superscript 𝑒 𝑧 modified-Bessel-first-kind ℓ 𝑠 ℎ superscript 𝑒 𝑧 {\displaystyle{\displaystyle\mathrm{Io}_{2m+1}\left(z,h\right)=(-1)^{s}\sum_{% \ell=0}^{\infty}(-1)^{\ell}\frac{A_{2\ell+1}^{2m+1}(h^{2})}{A_{2s+1}^{2m+1}(h^% {2})}\left(I_{\ell-s}\left(he^{-z}\right)I_{\ell+s+1}\left(he^{z}\right)-I_{% \ell+s+1}\left(he^{-z}\right)I_{\ell-s}\left(he^{z}\right)\right),}}
    0 references
    DLMF id
    DLMF:28.24.E9
    0 references
    Symbols used
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ahdec
    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.m2acdec
    0 references
    modified Mathieu function
    Defining formula
    Io n ⁡ ( z , h ) modified-Mathieu-Io 𝑛 𝑧 ℎ {\displaystyle{\displaystyle\mathrm{Io}_{\NVar{n}}\left(\NVar{z},\NVar{h}% \right)}}
    xml-id
    C28.S20.E18.m2aadec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1gdec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1hdec
    0 references
    Q12272
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1hdec
    0 references
    Q12287
    Defining formula
    A m ⁢ ( q ) subscript 𝐴 𝑚 𝑞 {\displaystyle{\displaystyle A_{m}(q)}}
    xml-id
    C28.S4.XMD1.m1cdec
    0 references
     
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