DLMF:28.10.E9 (Q8288)

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DLMF:28.10.E9
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    Statements

    Defining formula
    ∫ 0 π / 2 J 0 ⁡ ( 2 ⁢ q ⁢ ( cos 2 ⁡ τ - sin 2 ⁡ ζ ) ) ⁢ ce 2 ⁢ n ⁡ ( τ , q ) ⁢ d τ = w II ⁢ ( 1 2 ⁢ π ; a 2 ⁢ n ⁡ ( q ) , q ) ⁢ ce 2 ⁢ n ⁡ ( ζ , q ) , superscript subscript 0 𝜋 2 Bessel-J 0 2 𝑞 2 𝜏 2 𝜁 Mathieu-ce 2 𝑛 𝜏 𝑞 𝜏 subscript 𝑤 II 1 2 𝜋 Mathieu-eigenvalue-a 2 𝑛 𝑞 𝑞 Mathieu-ce 2 𝑛 𝜁 𝑞 {\displaystyle{\displaystyle\int_{0}^{\ifrac{\pi}{2}}J_{0}\left(2\sqrt{q({\cos% ^{2}}\tau-{\sin^{2}}\zeta)}\right)\mathrm{ce}_{2n}\left(\tau,q\right)\mathrm{d% }\tau=w_{\mbox{\tiny II}}(\tfrac{1}{2}\pi;a_{2n}\left(q\right),q)\mathrm{ce}_{% 2n}\left(\zeta,q\right),}}
    0 references
    DLMF id
    DLMF:28.10.E9
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2adec
    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.m7addec
    0 references
    Q12281
    Defining formula
    a n ⁡ ( q ) Mathieu-eigenvalue-a 𝑛 𝑞 {\displaystyle{\displaystyle a_{\NVar{n}}\left(\NVar{q}\right)}}
    xml-id
    C28.S2.SS5.p1.m3adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2afdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ahdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ahdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2agdec
    0 references
    Q12271
    Defining formula
    q = h 2 𝑞 superscript ℎ 2 {\displaystyle{\displaystyle q=h^{2}}}
    xml-id
    C28.S1.XMD10.m1dec
    0 references
    Q12279
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C28.S1.XMD2.m1hdec
    0 references
     
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