DLMF:28.28.E34 (Q8461): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: DLMF:28.28.E27 / rank
 
Normal rank
Property / Symbols used: DLMF:28.28.E27 / qualifier
 
Defining formula:

α ν , m ( s ) subscript superscript 𝛼 𝑠 𝜈 𝑚 {\displaystyle{\displaystyle\alpha^{(s)}_{\nu,m}}}

\alpha^{(s)}_{\nu,m}
Property / Symbols used: DLMF:28.28.E27 / qualifier
 
xml-id: C28.S28.XMD7.m1fdec

Latest revision as of 15:28, 2 January 2020

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DLMF:28.28.E34
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    Statements

    Defining formula
    1 π 2 ⁢ ⨍ 0 2 ⁢ π me ν ′ ⁡ ( t , h 2 ) ⁢ me - ν - 2 ⁢ m - 1 ⁡ ( t , h 2 ) sin ⁡ t ⁢ d t = ( - 1 ) m + 1 ⁢ i ⁢ h ⁢ α ν , m ( 0 ) ⁢ D 1 ⁡ ( ν , ν + 2 ⁢ m + 1 , 0 ) , 1 superscript 𝜋 2 superscript subscript principal-value-integral 0 2 𝜋 diffop Mathieu-me 𝜈 1 𝑡 superscript ℎ 2 Mathieu-me 𝜈 2 𝑚 1 𝑡 superscript ℎ 2 𝑡 𝑡 superscript 1 𝑚 1 imaginary-unit ℎ subscript superscript 𝛼 0 𝜈 𝑚 Mathieu-D 1 𝜈 𝜈 2 𝑚 1 0 {\displaystyle{\displaystyle\dfrac{1}{\pi^{2}}\pvint_{0}^{2\pi}\dfrac{\mathrm{% me}_{\nu}'\left(t,h^{2}\right)\mathrm{me}_{-\nu-2m-1}\left(t,h^{2}\right)}{% \sin t}\mathrm{d}t=(-1)^{m+1}\mathrm{i}h\alpha^{(0)}_{\nu,m}\mathrm{D}_{1}% \left(\nu,\nu+2m+1,0\right),}}
    0 references
    DLMF id
    DLMF:28.28.E34
    0 references
    Symbols used
    Q12317
    Defining formula
    D j ⁡ ( ν , μ , z ) Mathieu-D 𝑗 𝜈 𝜇 𝑧 {\displaystyle{\displaystyle\mathrm{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},% \NVar{z}\right)}}
    xml-id
    C28.S28.E24.m1ajdec
    0 references
    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.m2amdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aacdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaadec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2awdec
    0 references
    Cauchy principal value
    Defining formula
    ⨍ a b superscript subscript principal-value-integral 𝑎 𝑏 {\displaystyle{\displaystyle\pvint_{\NVar{a}}^{\NVar{b}}}}
    xml-id
    C1.S4.E24.m2adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2akdec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1odec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1acdec
    0 references
    Q12278
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C28.S1.XMD7.m1pdec
    0 references
    DLMF:28.28.E27
    Defining formula
    α ν , m ( s ) subscript superscript 𝛼 𝑠 𝜈 𝑚 {\displaystyle{\displaystyle\alpha^{(s)}_{\nu,m}}}
    xml-id
    C28.S28.XMD7.m1fdec
    0 references
     
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