DLMF:28.28.E4 (Q8425): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: radial Mathieu function / rank
 
Normal rank
Property / Symbols used: radial Mathieu function / qualifier
 
Defining formula:

Mc n ( j ) ⁡ ( z , h ) modified-Mathieu-Mc 𝑗 𝑛 𝑧 ℎ {\displaystyle{\displaystyle{\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z% },\NVar{h}\right)}}

\radMathieuMc{\NVar{j}}{\NVar{n}}@{\NVar{z}}{\NVar{h}}
Property / Symbols used: radial Mathieu function / qualifier
 
xml-id: C28.S20.E15.m2aadec
Property / Symbols used
 
Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / rank
 
Normal rank
Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / qualifier
 
Defining formula:

∂ ⁡ f ∂ ⁡ x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}

\pderiv{\NVar{f}}{\NVar{x}}
Property / Symbols used: partial derivative of $$f$$ with respect to $$x$$ / qualifier
 
xml-id: C1.S5.E3.m4adec
Property / Symbols used
 
Property / Symbols used: partial differential of $$x$$ / rank
 
Normal rank
Property / Symbols used: partial differential of $$x$$ / qualifier
 
Defining formula:

∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}

\pdiff{\NVar{x}}
Property / Symbols used: partial differential of $$x$$ / qualifier
 
xml-id: C1.S5.E3.m2adec
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.m1bdec
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.m1bdec
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.m1cdec
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.m1cdec
Property / Symbols used
 
Property / Symbols used: Q12311 / rank
 
Normal rank
Property / Symbols used: Q12311 / qualifier
 
Defining formula:

α 𝛼 {\displaystyle{\displaystyle\alpha}}

\alpha
Property / Symbols used: Q12311 / qualifier
 
xml-id: C28.S28.XMD1.m1cdec

Latest revision as of 15:17, 2 January 2020

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DLMF:28.28.E4
No description defined

    Statements

    Defining formula
    i ⁢ h π ⁢ ∫ 0 2 ⁢ π ∂ ⁡ w ∂ ⁡ α ⁢ e 2 ⁢ i ⁢ h ⁢ w ⁢ ce n ⁡ ( t , h 2 ) ⁢ d t = i n ⁢ ce n ′ ⁡ ( α , h 2 ) ⁢ Mc n ( 1 ) ⁡ ( z , h ) , imaginary-unit ℎ 𝜋 superscript subscript 0 2 𝜋 partial-derivative 𝑤 𝛼 superscript 𝑒 2 imaginary-unit ℎ 𝑤 Mathieu-ce 𝑛 𝑡 superscript ℎ 2 𝑡 imaginary-unit 𝑛 diffop Mathieu-ce 𝑛 1 𝛼 superscript ℎ 2 modified-Mathieu-Mc 1 𝑛 𝑧 ℎ {\displaystyle{\displaystyle\dfrac{\mathrm{i}h}{\pi}\int_{0}^{2\pi}\frac{% \partial w}{\partial\alpha}e^{2\mathrm{i}hw}\mathrm{ce}_{n}\left(t,h^{2}\right% )\mathrm{d}t={\mathrm{i}^{n}}\mathrm{ce}_{n}'\left(\alpha,h^{2}\right){\mathrm% {Mc}^{(1)}_{n}}\left(z,h\right),}}
    0 references
    DLMF id
    DLMF:28.28.E4
    0 references
    Symbols used
    Q12283
    Defining formula
    ce n ⁡ ( z , q ) Mathieu-ce 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S2.SS6.p1.m7aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
    0 references
    radial Mathieu function
    Defining formula
    Mc n ( j ) ⁡ ( z , h ) modified-Mathieu-Mc 𝑗 𝑛 𝑧 ℎ {\displaystyle{\displaystyle{\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z% },\NVar{h}\right)}}
    xml-id
    C28.S20.E15.m2aadec
    0 references
    partial derivative of $$f$$ with respect to $$x$$
    Defining formula
    ∂ ⁡ f ∂ ⁡ x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
    xml-id
    C1.S5.E3.m4adec
    0 references
    partial differential of $$x$$
    Defining formula
    ∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    xml-id
    C1.S5.E3.m2adec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1bdec
    0 references
    Q12279
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C28.S1.XMD2.m1bdec
    0 references
    Q12272
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1cdec
    0 references
    Q12274
    Defining formula
    w ⁢ ( z ) 𝑤 𝑧 {\displaystyle{\displaystyle w(z)}}
    xml-id
    C28.S2.XMD1.m1cdec
    0 references
    Q12311
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C28.S28.XMD1.m1cdec
    0 references
     
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