DLMF:28.20.E12 (Q8344)

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DLMF:28.20.E12
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    Statements

    Defining formula
    M ν ( 2 ) ⁡ ( z , h ) = Y ν ⁡ ( 2 ⁢ h ⁢ cosh ⁡ z ) + e | ℑ ⁡ ( 2 ⁢ h ⁢ cosh ⁡ z ) | ⁢ O ⁡ ( ( sech ⁡ z ) 3 / 2 ) , modified-Mathieu-M 2 𝜈 𝑧 ℎ Bessel-Y-Weber 𝜈 2 ℎ 𝑧 superscript 𝑒 2 ℎ 𝑧 Big-O superscript 𝑧 3 2 {\displaystyle{\displaystyle{\mathrm{M}^{(2)}_{\nu}}\left(z,h\right)=Y_{\nu}% \left(2h\cosh z\right)+e^{|\Im(2h\cosh z)|}O\left((\operatorname{sech}z)^{3/2}% \right),}}
    0 references
    DLMF id
    DLMF:28.20.E12
    0 references
    Symbols used
    Bessel function of the second kind
    Defining formula
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2adec
    0 references
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2aedec
    0 references
    hyperbolic secant function
    Defining formula
    sech ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{sech}\NVar{z}}}
    xml-id
    C4.S28.E6.m2acdec
    0 references
    Q10901
    Defining formula
    ℑ ⁡ absent {\displaystyle{\displaystyle\Im}}
    xml-id
    C1.S9.E2.m2aadec
    0 references
    Q12301
    Defining formula
    M ν ( j ) ⁡ ( z , h ) modified-Mathieu-M 𝑗 𝜈 𝑧 ℎ {\displaystyle{\displaystyle{\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{% z},\NVar{h}\right)}}
    xml-id
    C28.S20.SS3.p1.m4acdec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1ddec
    0 references
    Q12272
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1jdec
    0 references
     
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