DLMF:28.15.E3 (Q8327)

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DLMF:28.15.E3
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    Statements

    Defining formula
    me ν ⁡ ( z , q ) = e i ⁢ ν ⁢ z - q 4 ⁢ ( 1 ν + 1 ⁢ e i ⁢ ( ν + 2 ) ⁢ z - 1 ν - 1 ⁢ e i ⁢ ( ν - 2 ) ⁢ z ) + q 2 32 ⁢ ( 1 ( ν + 1 ) ⁢ ( ν + 2 ) ⁢ e i ⁢ ( ν + 4 ) ⁢ z + 1 ( ν - 1 ) ⁢ ( ν - 2 ) ⁢ e i ⁢ ( ν - 4 ) ⁢ z - 2 ⁢ ( ν 2 + 1 ) ( ν 2 - 1 ) 2 ⁢ e i ⁢ ν ⁢ z ) + ⋯ ; Mathieu-me 𝜈 𝑧 𝑞 superscript 𝑒 imaginary-unit 𝜈 𝑧 𝑞 4 1 𝜈 1 superscript 𝑒 imaginary-unit 𝜈 2 𝑧 1 𝜈 1 superscript 𝑒 imaginary-unit 𝜈 2 𝑧 superscript 𝑞 2 32 1 𝜈 1 𝜈 2 superscript 𝑒 imaginary-unit 𝜈 4 𝑧 1 𝜈 1 𝜈 2 superscript 𝑒 imaginary-unit 𝜈 4 𝑧 2 superscript 𝜈 2 1 superscript superscript 𝜈 2 1 2 superscript 𝑒 imaginary-unit 𝜈 𝑧 ⋯ {\displaystyle{\displaystyle\mathrm{me}_{\nu}\left(z,q\right)=e^{\mathrm{i}\nu z% }-\frac{q}{4}\left(\frac{1}{\nu+1}e^{\mathrm{i}(\nu+2)z}-\frac{1}{\nu-1}e^{% \mathrm{i}(\nu-2)z}\right)+\frac{q^{2}}{32}\left(\frac{1}{(\nu+1)(\nu+2)}e^{% \mathrm{i}(\nu+4)z}+\frac{1}{(\nu-1)(\nu-2)}e^{\mathrm{i}(\nu-4)z}-\frac{2(\nu% ^{2}+1)}{(\nu^{2}-1)^{2}}e^{\mathrm{i}\nu z}\right)+\cdots;}}
    0 references
    DLMF id
    DLMF:28.15.E3
    0 references
    Symbols used
    Q12299
    Defining formula
    me n ⁡ ( z , q ) Mathieu-me 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S12.SS2.p2.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q12271
    Defining formula
    q = h 2 𝑞 superscript ℎ 2 {\displaystyle{\displaystyle q=h^{2}}}
    xml-id
    C28.S1.XMD10.m1bdec
    0 references
    Q12272
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1dec
    0 references
    Q12278
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C28.S1.XMD7.m1bdec
    0 references
     
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