DLMF:28.28.E43 (Q8474): Difference between revisions

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Changed an Item: Add constraint
Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

sin z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2apdec

Revision as of 15:31, 2 January 2020

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DLMF:28.28.E43
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    Statements

    β ^ n , m = 1 2 π 0 2 π sin t se n ( t , h 2 ) ce m ( t , h 2 ) d t = ( - 1 ) p 2 i π se n ( 0 , h 2 ) ce m ( 0 , h 2 ) h Dsc 1 ( n , m , 0 ) . subscript ^ 𝛽 𝑛 𝑚 1 2 𝜋 superscript subscript 0 2 𝜋 𝑡 Mathieu-se 𝑛 𝑡 superscript 2 Mathieu-ce 𝑚 𝑡 superscript 2 𝑡 superscript 1 𝑝 2 imaginary-unit 𝜋 diffop Mathieu-se 𝑛 1 0 superscript 2 Mathieu-ce 𝑚 0 superscript 2 Mathieu-Dsc 1 𝑛 𝑚 0 {\displaystyle{\displaystyle\widehat{\beta}_{n,m}=\dfrac{1}{2\pi}\int_{0}^{2% \pi}\sin t\mathrm{se}_{n}\left(t,h^{2}\right)\mathrm{ce}_{m}\left(t,h^{2}% \right)\mathrm{d}t=(-1)^{p}\dfrac{2}{\mathrm{i}\pi}\dfrac{\mathrm{se}_{n}'% \left(0,h^{2}\right)\mathrm{ce}_{m}\left(0,h^{2}\right)}{h\mathrm{Dsc}_{1}% \left(n,m,0\right)}.}}
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    DLMF:28.28.E43
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    Dsc j ( n , m , z ) Mathieu-Dsc 𝑗 𝑛 𝑚 𝑧 {\displaystyle{\displaystyle\mathrm{Dsc}_{\NVar{j}}\left(\NVar{n},\NVar{m},% \NVar{z}\right)}}
    C28.S28.E40.m1acdec
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    ce n ( z , q ) Mathieu-ce 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    C28.S2.SS6.p1.m7aidec
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    se n ( z , q ) Mathieu-se 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    C28.S2.SS6.p1.m8ajdec
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    π {\displaystyle{\displaystyle\pi}}
    C3.S12.E1.m2aaidec
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    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    C1.S4.SS4.m1aagdec
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    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    C1.S9.E1.m2aacdec
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    {\displaystyle{\displaystyle\int}}
    C1.S4.SS4.m3aafdec
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    sin z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    C4.S14.E1.m2apdec
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