DLMF:28.28.E13 (Q8437): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: second solution, Mathieu’s equation / rank
 
Normal rank
Property / Symbols used: second solution, Mathieu’s equation / qualifier
 
Defining formula:

fe n ⁡ ( z , q ) Mathieu-fe 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}

\Mathieufe{\NVar{n}}@{\NVar{z}}{\NVar{q}}
Property / Symbols used: second solution, Mathieu’s equation / qualifier
 
xml-id: C28.S5.E1.m2adec
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.m2akdec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aidec
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.m2aidec
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.m2addec
Property / Symbols used
 
Property / Symbols used: hyperbolic sine function / rank
 
Normal rank
Property / Symbols used: hyperbolic sine function / qualifier
 
Defining formula:

sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}

\sinh@@{\NVar{z}}
Property / Symbols used: hyperbolic sine function / qualifier
 
xml-id: C4.S28.E1.m2abdec
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.m2ajdec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aidec
Property / Symbols used
 
Property / Symbols used: modified Mathieu function / rank
 
Normal rank
Property / Symbols used: modified Mathieu function / qualifier
 
Defining formula:

Fe n ⁡ ( z , q ) modified-Mathieu-Fe 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{Fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}

\modMathieuFe{\NVar{n}}@{\NVar{z}}{\NVar{q}}
Property / Symbols used: modified Mathieu function / qualifier
 
xml-id: C28.S20.E6.m2adec
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.m2abdec
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.m1jdec
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.m1kdec

Latest revision as of 15:21, 2 January 2020

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English
DLMF:28.28.E13
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ e 2 ⁢ i ⁢ h ⁢ cosh ⁡ z ⁢ cosh ⁡ t ⁢ sinh ⁡ z ⁢ sinh ⁡ t ⁢ Fe m ⁡ ( t , h 2 ) ⁢ d t = - π 4 ⁢ h ⁢ i m ⁢ fe m ′ ⁡ ( 0 , h 2 ) ⁢ Mc m ( 3 ) ⁡ ( z , h ) , superscript subscript 0 superscript 𝑒 2 imaginary-unit ℎ 𝑧 𝑡 𝑧 𝑡 modified-Mathieu-Fe 𝑚 𝑡 superscript ℎ 2 𝑡 𝜋 4 ℎ imaginary-unit 𝑚 diffop Mathieu-fe 𝑚 1 0 superscript ℎ 2 modified-Mathieu-Mc 3 𝑚 𝑧 ℎ {\displaystyle{\displaystyle\int_{0}^{\infty}e^{2\mathrm{i}h\cosh z\cosh t}% \sinh z\sinh t\mathrm{Fe}_{m}\left(t,h^{2}\right)\mathrm{d}t=-\dfrac{\pi}{4h}{% \mathrm{i}^{m}}\mathrm{fe}_{m}'\left(0,h^{2}\right){\mathrm{Mc}^{(3)}_{m}}% \left(z,h\right),}}
    0 references
    DLMF id
    DLMF:28.28.E13
    0 references
    Symbols used
    second solution, Mathieu’s equation
    Defining formula
    fe n ⁡ ( z , q ) Mathieu-fe 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S5.E1.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2akdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aidec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aidec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2addec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ajdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aidec
    0 references
    modified Mathieu function
    Defining formula
    Fe n ⁡ ( z , q ) modified-Mathieu-Fe 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{Fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S20.E6.m2adec
    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.m2abdec
    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.m1jdec
    0 references
    Q12272
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1kdec
    0 references
     
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