DLMF:28.24.E4 (Q8401): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
(7 intermediate revisions by the same user not shown)
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2acdec
Property / Symbols used
 
Property / Symbols used: radial Mathieu function / rank
 
Normal rank
Property / Symbols used: radial Mathieu function / qualifier
 
Defining formula:

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

\radMathieuMs{\NVar{j}}{\NVar{n}}@{\NVar{z}}{\NVar{h}}
Property / Symbols used: radial Mathieu function / qualifier
 
xml-id: C28.S20.E16.m2adec
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.m1bdec
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.m1cdec
Property / Symbols used
 
Property / Symbols used: Q12291 / rank
 
Normal rank
Property / Symbols used: Q12291 / qualifier
 
Defining formula:

j 𝑗 {\displaystyle{\displaystyle j}}

j
Property / Symbols used: Q12291 / qualifier
 
xml-id: C28.S1.XMD3.m1cdec
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.m1cdec
Property / Symbols used
 
Property / Symbols used: Q12304 / rank
 
Normal rank
Property / Symbols used: Q12304 / qualifier
 
Defining formula:

𝒞 μ ( j ) superscript subscript 𝒞 𝜇 𝑗 {\displaystyle{\displaystyle\mathcal{C}_{\mu}^{(j)}}}

\mathcal{C}_{\mu}^{(j)}
Property / Symbols used: Q12304 / qualifier
 
xml-id: C28.S23.XMD1.m1cdec
Property / Symbols used
 
Property / Symbols used: Q12288 / rank
 
Normal rank
Property / Symbols used: Q12288 / qualifier
 
Defining formula:

B m ⁢ ( q ) subscript 𝐵 𝑚 𝑞 {\displaystyle{\displaystyle B_{m}(q)}}

B_{m}(q)
Property / Symbols used: Q12288 / qualifier
 
xml-id: C28.S4.XMD2.m1dec

Latest revision as of 15:11, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:28.24.E4
No description defined

    Statements

    Defining formula
    Ms 2 ⁢ m + 1 ( j ) ⁡ ( z , h ) = ( - 1 ) m ⁢ ∑ ℓ = 0 ∞ ( - 1 ) ℓ ⁢ B 2 ⁢ ℓ + 1 2 ⁢ m + 1 ⁢ ( h 2 ) B 2 ⁢ s + 1 2 ⁢ m + 1 ⁢ ( h 2 ) ⁢ ( J ℓ - s ⁡ ( h ⁢ e - z ) ⁢ 𝒞 ℓ + s + 1 ( j ) ⁢ ( h ⁢ e z ) - J ℓ + s + 1 ⁡ ( h ⁢ e - z ) ⁢ 𝒞 ℓ - s ( j ) ⁢ ( h ⁢ e z ) ) , modified-Mathieu-Ms 𝑗 2 𝑚 1 𝑧 ℎ superscript 1 𝑚 superscript subscript ℓ 0 superscript 1 ℓ superscript subscript 𝐵 2 ℓ 1 2 𝑚 1 superscript ℎ 2 superscript subscript 𝐵 2 𝑠 1 2 𝑚 1 superscript ℎ 2 Bessel-J ℓ 𝑠 ℎ superscript 𝑒 𝑧 superscript subscript 𝒞 ℓ 𝑠 1 𝑗 ℎ superscript 𝑒 𝑧 Bessel-J ℓ 𝑠 1 ℎ superscript 𝑒 𝑧 superscript subscript 𝒞 ℓ 𝑠 𝑗 ℎ superscript 𝑒 𝑧 {\displaystyle{\displaystyle{\mathrm{Ms}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m% }\sum_{\ell=0}^{\infty}(-1)^{\ell}\frac{B_{2\ell+1}^{2m+1}(h^{2})}{B_{2s+1}^{2% m+1}(h^{2})}\left(J_{\ell-s}\left(he^{-z}\right){\cal C}_{\ell+s+1}^{(j)}(he^{% z})-J_{\ell+s+1}\left(he^{-z}\right){\cal C}_{\ell-s}^{(j)}(he^{z})\right),}}
    0 references
    DLMF id
    DLMF:28.24.E4
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    radial Mathieu function
    Defining formula
    Ms n ( j ) ⁡ ( z , h ) modified-Mathieu-Ms 𝑗 𝑛 𝑧 ℎ {\displaystyle{\displaystyle{\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z% },\NVar{h}\right)}}
    xml-id
    C28.S20.E16.m2adec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1bdec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1cdec
    0 references
    Q12291
    Defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C28.S1.XMD3.m1cdec
    0 references
    Q12272
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1cdec
    0 references
    Q12304
    Defining formula
    𝒞 μ ( j ) superscript subscript 𝒞 𝜇 𝑗 {\displaystyle{\displaystyle\mathcal{C}_{\mu}^{(j)}}}
    xml-id
    C28.S23.XMD1.m1cdec
    0 references
    Q12288
    Defining formula
    B m ⁢ ( q ) subscript 𝐵 𝑚 𝑞 {\displaystyle{\displaystyle B_{m}(q)}}
    xml-id
    C28.S4.XMD2.m1dec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q8401&oldid=55228"