DLMF:28.4.E25 (Q8219): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: order not exceeding / rank
 
Normal rank
Property / Symbols used: order not exceeding / qualifier
 
Defining formula:

O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}

\bigO@{\NVar{x}}
Property / Symbols used: order not exceeding / qualifier
 
xml-id: C2.S1.E3.m2addec
Property / Symbols used
 
Property / Symbols used: Q12281 / rank
 
Normal rank
Property / Symbols used: Q12281 / qualifier
 
Defining formula:

a n ⁡ ( q ) Mathieu-eigenvalue-a 𝑛 𝑞 {\displaystyle{\displaystyle a_{\NVar{n}}\left(\NVar{q}\right)}}

\Mathieueigvala{\NVar{n}}@{\NVar{q}}
Property / Symbols used: Q12281 / qualifier
 
xml-id: C28.S2.SS5.p1.m3acdec
Property / Symbols used
 
Property / Symbols used: Q10759 / rank
 
Normal rank
Property / Symbols used: Q10759 / qualifier
 
Defining formula:

( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}

\Pochhammersym{\NVar{a}}{\NVar{n}}
Property / Symbols used: Q10759 / qualifier
 
xml-id: C5.S2.SS3.m1adec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: Q12284 / rank
 
Normal rank
Property / Symbols used: Q12284 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q12284 / qualifier
 
xml-id: C28.S1.XMD1.m1wdec
Property / Symbols used
 
Property / Symbols used: Q12271 / rank
 
Normal rank
Property / Symbols used: Q12271 / qualifier
 
Defining formula:

q = h 2 𝑞 superscript ℎ 2 {\displaystyle{\displaystyle q=h^{2}}}

q=h^{2}
Property / Symbols used: Q12271 / qualifier
 
xml-id: C28.S1.XMD10.m1tdec
Property / Symbols used
 
Property / Symbols used: Q12279 / rank
 
Normal rank
Property / Symbols used: Q12279 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q12279 / qualifier
 
xml-id: C28.S1.XMD2.m1udec
Property / Symbols used
 
Property / Symbols used: Q12274 / rank
 
Normal rank
Property / Symbols used: Q12274 / qualifier
 
Defining formula:

w ⁢ ( z ) 𝑤 𝑧 {\displaystyle{\displaystyle w(z)}}

w(z)
Property / Symbols used: Q12274 / qualifier
 
xml-id: C28.S2.XMD1.m1dec
Property / Symbols used
 
Property / Symbols used: Q12287 / rank
 
Normal rank
Property / Symbols used: Q12287 / qualifier
 
Defining formula:

A m ⁢ ( q ) subscript 𝐴 𝑚 𝑞 {\displaystyle{\displaystyle A_{m}(q)}}

A_{m}(q)
Property / Symbols used: Q12287 / qualifier
 
xml-id: C28.S4.XMD1.m1odec

Latest revision as of 14:42, 2 January 2020

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DLMF:28.4.E25
No description defined

    Statements

    Defining formula
    A 2 ⁢ m + 1 2 ⁢ n + 1 ⁢ ( q ) A 1 2 ⁢ n + 1 ⁢ ( q ) = ( - 1 ) m + 1 ( ( 1 2 ) m + 1 ) 2 ⁢ ( q 4 ) m + 1 ⁢ 2 ⁢ ( 1 + O ⁡ ( m - 1 ) ) w II ′ ⁢ ( 1 2 ⁢ π ; a 2 ⁢ n + 1 ⁡ ( q ) , q ) , subscript superscript 𝐴 2 𝑛 1 2 𝑚 1 𝑞 subscript superscript 𝐴 2 𝑛 1 1 𝑞 superscript 1 𝑚 1 superscript Pochhammer 1 2 𝑚 1 2 superscript 𝑞 4 𝑚 1 2 1 Big-O superscript 𝑚 1 subscript superscript 𝑤 ′ II 1 2 𝜋 Mathieu-eigenvalue-a 2 𝑛 1 𝑞 𝑞 {\displaystyle{\displaystyle\frac{A^{2n+1}_{2m+1}(q)}{A^{2n+1}_{1}(q)}=\frac{(% -1)^{m+1}}{\left({\left(\tfrac{1}{2}\right)_{m+1}}\right)^{2}}\left(\frac{q}{4% }\right)^{m+1}\frac{2\left(1+O\left(m^{-1}\right)\right)}{w^{\prime}_{\mbox{% \tiny II}}(\frac{1}{2}\pi;a_{2n+1}\left(q\right),q)},}}
    0 references
    DLMF id
    DLMF:28.4.E25
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2addec
    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.m3acdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1wdec
    0 references
    Q12271
    Defining formula
    q = h 2 𝑞 superscript ℎ 2 {\displaystyle{\displaystyle q=h^{2}}}
    xml-id
    C28.S1.XMD10.m1tdec
    0 references
    Q12279
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C28.S1.XMD2.m1udec
    0 references
    Q12274
    Defining formula
    w ⁢ ( z ) 𝑤 𝑧 {\displaystyle{\displaystyle w(z)}}
    xml-id
    C28.S2.XMD1.m1dec
    0 references
    Q12287
    Defining formula
    A m ⁢ ( q ) subscript 𝐴 𝑚 𝑞 {\displaystyle{\displaystyle A_{m}(q)}}
    xml-id
    C28.S4.XMD1.m1odec
    0 references
     
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