DLMF:28.25.E1 (Q8411): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10931 / rank
 
Normal rank
Property / Symbols used: Q10931 / qualifier
 
Defining formula:

∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}

\asympexp
Property / Symbols used: Q10931 / qualifier
 
xml-id: C2.S1.SS3.p1.m11adec
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.m2adec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2adec
Property / Symbols used
 
Property / Symbols used: hyperbolic cosine function / rank
 
Normal rank
Property / Symbols used: hyperbolic cosine function / qualifier
 
Defining formula:

cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}

\cosh@@{\NVar{z}}
Property / Symbols used: hyperbolic cosine function / qualifier
 
xml-id: C4.S28.E2.m2adec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2adec
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.m4adec
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.m1dec
Property / Symbols used
 
Property / Symbols used: Q12294 / rank
 
Normal rank
Property / Symbols used: Q12294 / qualifier
 
Defining formula:

h ℎ {\displaystyle{\displaystyle h}}

h
Property / Symbols used: Q12294 / qualifier
 
xml-id: C28.S1.XMD11.m1dec
Property / Symbols used
 
Property / Symbols used: Q12272 / rank
 
Normal rank
Property / Symbols used: Q12272 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q12272 / qualifier
 
xml-id: C28.S1.XMD6.m1dec
Property / Symbols used
 
Property / Symbols used: Q12278 / rank
 
Normal rank
Property / Symbols used: Q12278 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q12278 / qualifier
 
xml-id: C28.S1.XMD7.m1dec
Property / Symbols used
 
Property / Symbols used: Q12308 / rank
 
Normal rank
Property / Symbols used: Q12308 / qualifier
 
Defining formula:

D m ± superscript subscript 𝐷 𝑚 plus-or-minus {\displaystyle{\displaystyle D_{m}^{\pm}}}

D_{m}^{\pm}
Property / Symbols used: Q12308 / qualifier
 
xml-id: C28.S25.XMD1.m1dec

Latest revision as of 15:14, 2 January 2020

No description defined
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English
DLMF:28.25.E1
No description defined

    Statements

    Defining formula
    M ν ( 3 , 4 ) ⁡ ( z , h ) ∼ e ± i ⁢ ( 2 ⁢ h ⁢ cosh ⁡ z - ( 1 2 ⁢ ν + 1 4 ) ⁢ π ) ( π ⁢ h ⁢ ( cosh ⁡ z + 1 ) ) 1 2 ⁢ ∑ m = 0 ∞ D m ± ( ∓ 4 ⁢ i ⁢ h ⁢ ( cosh ⁡ z + 1 ) ) m , asymptotic-expansion modified-Mathieu-M 3 4 𝜈 𝑧 ℎ superscript 𝑒 plus-or-minus imaginary-unit 2 ℎ 𝑧 1 2 𝜈 1 4 𝜋 superscript 𝜋 ℎ 𝑧 1 1 2 superscript subscript 𝑚 0 subscript superscript 𝐷 plus-or-minus 𝑚 superscript minus-or-plus 4 imaginary-unit ℎ 𝑧 1 𝑚 {\displaystyle{\displaystyle{\mathrm{M}^{(3,4)}_{\nu}}\left(z,h\right)\sim% \frac{e^{\pm\mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu+\frac{1}{4}\right)% \pi\right)}}{\left(\pi h(\cosh z+1)\right)^{\frac{1}{2}}}\*\sum_{m=0}^{\infty}% \dfrac{D^{\pm}_{m}}{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)^{m}},}}
    0 references
    DLMF id
    DLMF:28.25.E1
    0 references
    Symbols used
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11adec
    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
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.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.m4adec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1dec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1dec
    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.m1dec
    0 references
    Q12308
    Defining formula
    D m ± superscript subscript 𝐷 𝑚 plus-or-minus {\displaystyle{\displaystyle D_{m}^{\pm}}}
    xml-id
    C28.S25.XMD1.m1dec
    0 references
     
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