DLMF:28.23.E3 (Q8387): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q12301 / rank
 
Normal rank
Property / Symbols used: Q12301 / qualifier
 
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)}}

\modMathieuM{\NVar{j}}{\NVar{\nu}}@{\NVar{z}}{\NVar{h}}
Property / Symbols used: Q12301 / qualifier
 
xml-id: C28.S20.SS3.p1.m4aadec

Revision as of 15:06, 2 January 2020

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DLMF:28.23.E3
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    Statements

    Defining formula
    me ν ′ ⁡ ( 0 , h 2 ) ⁢ M ν ( j ) ⁡ ( z , h ) = i ⁢ tanh ⁡ z ⁢ ∑ n = - ∞ ∞ ( - 1 ) n ⁢ ( ν + 2 ⁢ n ) ⁢ c 2 ⁢ n ν ⁢ ( h 2 ) ⁢ 𝒞 ν + 2 ⁢ n ( j ) ⁢ ( 2 ⁢ h ⁢ cosh ⁡ z ) , diffop Mathieu-me 𝜈 1 0 superscript ℎ 2 modified-Mathieu-M 𝑗 𝜈 𝑧 ℎ imaginary-unit 𝑧 superscript subscript 𝑛 superscript 1 𝑛 𝜈 2 𝑛 superscript subscript 𝑐 2 𝑛 𝜈 superscript ℎ 2 superscript subscript 𝒞 𝜈 2 𝑛 𝑗 2 ℎ 𝑧 {\displaystyle{\displaystyle\mathrm{me}_{\nu}'\left(0,h^{2}\right){\mathrm{M}^% {(j)}_{\nu}}\left(z,h\right)=\mathrm{i}\tanh z\sum_{n=-\infty}^{\infty}(-1)^{n% }(\nu+2n)c_{2n}^{\nu}(h^{2}){\cal C}_{\nu+2n}^{(j)}(2h\cosh z),}}
    0 references
    DLMF id
    DLMF:28.23.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.m2aadec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2aadec
    0 references
    hyperbolic tangent function
    Defining formula
    tanh ⁡ z 𝑧 {\displaystyle{\displaystyle\tanh\NVar{z}}}
    xml-id
    C4.S28.E4.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    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.m4aadec
    0 references
     
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