DLMF:28.28.E9 (Q8433): Difference between revisions

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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.m4abdec
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.m1fdec
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.m1gdec
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.m1bdec
Property / Symbols used
 
Property / Symbols used: Q12274 / rank
 
Normal rank
Property / Symbols used: Q12274 / qualifier
 
Defining formula:

w ⁢ ( z ) 𝑤 𝑧 {\displaystyle{\displaystyle w(z)}}

w(z)
Property / Symbols used: Q12274 / qualifier
 
xml-id: C28.S2.XMD1.m1gdec
Property / Symbols used
 
Property / Symbols used: Q12311 / rank
 
Normal rank
Property / Symbols used: Q12311 / qualifier
 
Defining formula:

α 𝛼 {\displaystyle{\displaystyle\alpha}}

\alpha
Property / Symbols used: Q12311 / qualifier
 
xml-id: C28.S28.XMD1.m1gdec
Property / Symbols used
 
Property / Symbols used: Q12312 / rank
 
Normal rank
Property / Symbols used: Q12312 / qualifier
 
Defining formula:

ℒ j subscript ℒ 𝑗 {\displaystyle{\displaystyle\mathcal{L}_{j}}}

\mathcal{L}_{j}
Property / Symbols used: Q12312 / qualifier
 
xml-id: C28.S28.XMD2.m1cdec

Latest revision as of 15:19, 2 January 2020

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DLMF:28.28.E9
No description defined

    Statements

    Defining formula
    1 2 ⁢ π ⁢ ∫ ℒ 1 e 2 ⁢ i ⁢ h ⁢ w ⁢ me ν ⁡ ( t , h 2 ) ⁢ d t = e i ⁢ ν ⁢ π / 2 ⁢ me ν ⁡ ( α , h 2 ) ⁢ M ν ( 1 ) ⁡ ( z , h ) . 1 2 𝜋 subscript subscript ℒ 1 superscript 𝑒 2 imaginary-unit ℎ 𝑤 Mathieu-me 𝜈 𝑡 superscript ℎ 2 𝑡 superscript 𝑒 imaginary-unit 𝜈 𝜋 2 Mathieu-me 𝜈 𝛼 superscript ℎ 2 modified-Mathieu-M 1 𝜈 𝑧 ℎ {\displaystyle{\displaystyle\dfrac{1}{2\pi}\int_{\mathcal{L}_{1}}e^{2\mathrm{i% }hw}\mathrm{me}_{\nu}\left(t,h^{2}\right)\mathrm{d}t=e^{\mathrm{i}\nu\pi/{2}}% \mathrm{me}_{\nu}\left(\alpha,h^{2}\right){\mathrm{M}^{(1)}_{\nu}}\left(z,h% \right).}}
    0 references
    DLMF id
    DLMF:28.28.E9
    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.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2agdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
    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.m4abdec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1fdec
    0 references
    Q12272
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1gdec
    0 references
    Q12278
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C28.S1.XMD7.m1bdec
    0 references
    Q12274
    Defining formula
    w ⁢ ( z ) 𝑤 𝑧 {\displaystyle{\displaystyle w(z)}}
    xml-id
    C28.S2.XMD1.m1gdec
    0 references
    Q12311
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C28.S28.XMD1.m1gdec
    0 references
    Q12312
    Defining formula
    ℒ j subscript ℒ 𝑗 {\displaystyle{\displaystyle\mathcal{L}_{j}}}
    xml-id
    C28.S28.XMD2.m1cdec
    0 references
     
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