DLMF:28.28.E38 (Q8467)

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DLMF:28.28.E38
No description defined

    Statements

    Defining formula
    α ^ n , m ( s ) = 1 2 ⁢ π ⁢ ∫ 0 2 ⁢ π cos ⁡ t ⁢ se n ⁡ ( t , h 2 ) ⁢ se m ⁡ ( t , h 2 ) ⁢ d t = ( - 1 ) p ⁢ 2 i ⁢ π ⁢ se n ′ ⁡ ( 0 , h 2 ) ⁢ se m ′ ⁡ ( 0 , h 2 ) h ⁢ Ds 2 ⁡ ( n , m , 0 ) . superscript subscript ^ 𝛼 𝑛 𝑚 𝑠 1 2 𝜋 superscript subscript 0 2 𝜋 𝑡 Mathieu-se 𝑛 𝑡 superscript ℎ 2 Mathieu-se 𝑚 𝑡 superscript ℎ 2 𝑡 superscript 1 𝑝 2 imaginary-unit 𝜋 diffop Mathieu-se 𝑛 1 0 superscript ℎ 2 diffop Mathieu-se 𝑚 1 0 superscript ℎ 2 ℎ Mathieu-Ds 2 𝑛 𝑚 0 {\displaystyle{\displaystyle\widehat{\alpha}_{n,m}^{(s)}=\dfrac{1}{2\pi}\int_{% 0}^{2\pi}\cos t\mathrm{se}_{n}\left(t,h^{2}\right)\mathrm{se}_{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{se}_{m}'\left(0,h^{2}\right)}{h\mathrm{Ds}_{2}% \left(n,m,0\right)}.}}
    0 references
    DLMF id
    DLMF:28.28.E38
    0 references
    Symbols used
    Q12318
    Defining formula
    Ds j ⁡ ( n , m , z ) Mathieu-Ds 𝑗 𝑛 𝑚 𝑧 {\displaystyle{\displaystyle\mathrm{Ds}_{\NVar{j}}\left(\NVar{n},\NVar{m},% \NVar{z}\right)}}
    xml-id
    C28.S28.E35.m1acdec
    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.m8agdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aafdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2akdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaddec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2azdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aacdec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1sdec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1agdec
    0 references
    Q12279
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C28.S1.XMD2.m1idec
    0 references
    DLMF:28.28.E49
    Defining formula
    α ^ n , m subscript ^ 𝛼 𝑛 𝑚 {\displaystyle{\displaystyle\widehat{\alpha}_{n,m}}}
    xml-id
    C28.S28.XMD11.m1bdec
    0 references
     
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