DLMF:28.28.E17 (Q8441): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2aldec
Property / Symbols used
 
Property / Symbols used: Q10818 / rank
 
Normal rank
Property / Symbols used: Q10818 / qualifier
 
Defining formula:

ℤ {\displaystyle{\displaystyle\mathbb{Z}}}

\Integers
Property / Symbols used: Q10818 / qualifier
 
xml-id: introduction.Sx4.p2.t1.r20.m2adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3amdec
Property / Symbols used
 
Property / Symbols used: Q12301 / rank
 
Normal rank
Property / Symbols used: Q12301 / qualifier
 
Defining formula:

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

\modMathieuM{\NVar{j}}{\NVar{\nu}}@{\NVar{z}}{\NVar{h}}
Property / Symbols used: Q12301 / qualifier
 
xml-id: C28.S20.SS3.p1.m4aedec
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.m1ndec
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.m1mdec
Property / Symbols used
 
Property / Symbols used: Q12278 / rank
 
Normal rank
Property / Symbols used: Q12278 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q12278 / qualifier
 
xml-id: C28.S1.XMD7.m1edec
Property / Symbols used
 
Property / Symbols used: Q12300 / rank
 
Normal rank
Property / Symbols used: Q12300 / qualifier
 
Defining formula:

c 2 ⁢ m ⁢ ( q ) subscript 𝑐 2 𝑚 𝑞 {\displaystyle{\displaystyle c_{2m}(q)}}

c_{2m}(q)
Property / Symbols used: Q12300 / qualifier
 
xml-id: C28.S14.XMD1.m1dec
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.m1dec
Property / Symbols used
 
Property / Symbols used: Q12314 / rank
 
Normal rank
Property / Symbols used: Q12314 / qualifier
 
Defining formula:

ϕ ⁢ ( z , t ) italic-ϕ 𝑧 𝑡 {\displaystyle{\displaystyle\phi(z,t)}}

\phi(z,t)
Property / Symbols used: Q12314 / qualifier
 
xml-id: C28.S28.XMD4.m1dec
Property / Symbols used
 
Property / Symbols used: Q12315 / rank
 
Normal rank
Property / Symbols used: Q12315 / qualifier
 
Defining formula:

R ⁢ ( z , t ) 𝑅 𝑧 𝑡 {\displaystyle{\displaystyle R(z,t)}}

R(z,t)
Property / Symbols used: Q12315 / qualifier
 
xml-id: C28.S28.XMD5.m1dec

Latest revision as of 15:22, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:28.28.E17
No description defined

    Statements

    Defining formula
    1 π ⁢ ∫ 0 π 𝒞 ν + 2 ⁢ s ( j ) ⁢ ( 2 ⁢ h ⁢ R ) ⁢ e - i ⁢ ( ν + 2 ⁢ s ) ⁢ ϕ ⁢ me ν ⁡ ( t , h 2 ) ⁢ d t = ( - 1 ) s ⁢ c 2 ⁢ s ν ⁢ ( h 2 ) ⁢ M ν ( j ) ⁡ ( z , h ) , 1 𝜋 superscript subscript 0 𝜋 subscript superscript 𝒞 𝑗 𝜈 2 𝑠 2 ℎ 𝑅 superscript 𝑒 imaginary-unit 𝜈 2 𝑠 italic-ϕ Mathieu-me 𝜈 𝑡 superscript ℎ 2 𝑡 superscript 1 𝑠 subscript superscript 𝑐 𝜈 2 𝑠 superscript ℎ 2 modified-Mathieu-M 𝑗 𝜈 𝑧 ℎ {\displaystyle{\displaystyle\dfrac{1}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{\nu% +2s}(2hR)e^{-\mathrm{i}(\nu+2s)\phi}\mathrm{me}_{\nu}\left(t,h^{2}\right)% \mathrm{d}t=(-1)^{s}c^{\nu}_{2s}(h^{2}){\mathrm{M}^{(j)}_{\nu}}\left(z,h\right% ),}}
    0 references
    DLMF id
    DLMF:28.28.E17
    0 references
    constraint
    s ∈ ℤ 𝑠 {\displaystyle{\displaystyle s\in\mathbb{Z}}}
    0 references
    Symbols used
    Q12299
    Defining formula
    me n ⁡ ( z , q ) Mathieu-me 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S12.SS2.p2.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aodec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1amdec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2akdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aldec
    0 references
    Q10818
    Defining formula
    ℤ {\displaystyle{\displaystyle\mathbb{Z}}}
    xml-id
    introduction.Sx4.p2.t1.r20.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3amdec
    0 references
    Q12301
    Defining formula
    M ν ( j ) ⁡ ( z , h ) modified-Mathieu-M 𝑗 𝜈 𝑧 ℎ {\displaystyle{\displaystyle{\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{% z},\NVar{h}\right)}}
    xml-id
    C28.S20.SS3.p1.m4aedec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1ndec
    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.m1mdec
    0 references
    Q12278
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C28.S1.XMD7.m1edec
    0 references
    Q12300
    Defining formula
    c 2 ⁢ m ⁢ ( q ) subscript 𝑐 2 𝑚 𝑞 {\displaystyle{\displaystyle c_{2m}(q)}}
    xml-id
    C28.S14.XMD1.m1dec
    0 references
    Q12304
    Defining formula
    𝒞 μ ( j ) superscript subscript 𝒞 𝜇 𝑗 {\displaystyle{\displaystyle\mathcal{C}_{\mu}^{(j)}}}
    xml-id
    C28.S23.XMD1.m1dec
    0 references
    Q12314
    Defining formula
    ϕ ⁢ ( z , t ) italic-ϕ 𝑧 𝑡 {\displaystyle{\displaystyle\phi(z,t)}}
    xml-id
    C28.S28.XMD4.m1dec
    0 references
    Q12315
    Defining formula
    R ⁢ ( z , t ) 𝑅 𝑧 𝑡 {\displaystyle{\displaystyle R(z,t)}}
    xml-id
    C28.S28.XMD5.m1dec
    0 references
     
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