DLMF:28.8.E2 (Q8266)

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DLMF:28.8.E2
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    Statements

    Defining formula
    b m + 1 ⁡ ( h 2 ) - a m ⁡ ( h 2 ) = 2 4 ⁢ m + 5 m ! ⁢ ( 2 π ) 1 / 2 ⁢ h m + ( 3 / 2 ) ⁢ e - 4 ⁢ h ⁢ ( 1 - 6 ⁢ m 2 + 14 ⁢ m + 7 32 ⁢ h + O ⁡ ( 1 h 2 ) ) . Mathieu-eigenvalue-b 𝑚 1 superscript ℎ 2 Mathieu-eigenvalue-a 𝑚 superscript ℎ 2 superscript 2 4 𝑚 5 𝑚 superscript 2 𝜋 1 2 superscript ℎ 𝑚 3 2 superscript 𝑒 4 ℎ 1 6 superscript 𝑚 2 14 𝑚 7 32 ℎ Big-O 1 superscript ℎ 2 {\displaystyle{\displaystyle b_{m+1}\left(h^{2}\right)-a_{m}\left(h^{2}\right)% =\frac{2^{4m+5}}{m!}\left(\frac{2}{\pi}\right)^{\ifrac{1}{2}}h^{m+(\ifrac{3}{2% })}e^{-4h}\*{\left(1-\frac{6m^{2}+14m+7}{32h}+O\left(\frac{1}{h^{2}}\right)% \right)}.}}
    0 references
    DLMF id
    DLMF:28.8.E2
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    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2adec
    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.m3aadec
    0 references
    Q12282
    Defining formula
    b n ⁡ ( q ) Mathieu-eigenvalue-b 𝑛 𝑞 {\displaystyle{\displaystyle b_{\NVar{n}}\left(\NVar{q}\right)}}
    xml-id
    C28.S2.SS5.p1.m4aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1adec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1adec
    0 references
     
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