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

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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

    Defining formula
    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 ℎ 2 Mathieu-ce 𝑚 𝑡 superscript ℎ 2 𝑡 superscript 1 𝑝 2 imaginary-unit 𝜋 diffop Mathieu-se 𝑛 1 0 superscript ℎ 2 Mathieu-ce 𝑚 0 superscript ℎ 2 ℎ 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)}.}}
    0 references
    DLMF id
    DLMF:28.28.E43
    0 references
    Symbols used
    Q12319
    Defining formula
    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)}}
    xml-id
    C28.S28.E40.m1acdec
    0 references
    Q12283
    Defining formula
    ce n ⁡ ( z , q ) Mathieu-ce 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S2.SS6.p1.m7aidec
    0 references
    Q12286
    Defining formula
    se n ⁡ ( z , q ) Mathieu-se 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S2.SS6.p1.m8ajdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aaidec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aagdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aacdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aafdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2apdec
    0 references
     
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