DLMF:18.17.E19 (Q5760): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older edit
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
(5 intermediate revisions by the same user not shown)
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.m2addec
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.m2aadec
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.m3ardec
Property / Symbols used
 
Property / Symbols used: Q11743 / rank
 
Normal rank
Property / Symbols used: Q11743 / qualifier
 
Defining formula:

y 𝑦 {\displaystyle{\displaystyle y}}

y
Property / Symbols used: Q11743 / qualifier
 
xml-id: C18.S1.XMD1.m1ndec
Property / Symbols used
 
Property / Symbols used: Q11726 / rank
 
Normal rank
Property / Symbols used: Q11726 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11726 / qualifier
 
xml-id: C18.S1.XMD6.m1rdec
Property / Symbols used
 
Property / Symbols used: Q11727 / rank
 
Normal rank
Property / Symbols used: Q11727 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11727 / qualifier
 
xml-id: C18.S2.XMD3.m1pdec

Latest revision as of 15:48, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:18.17.E19
No description defined

    Statements

    Defining formula
    ∫ - 1 1 P n ⁡ ( x ) ⁢ e i ⁢ x ⁢ y ⁢ d x = i n ⁢ 2 ⁢ π y ⁢ J n + 1 2 ⁡ ( y ) , superscript subscript 1 1 Legendre-spherical-polynomial 𝑛 𝑥 superscript 𝑒 𝑖 𝑥 𝑦 𝑥 superscript 𝑖 𝑛 2 𝜋 𝑦 Bessel-J 𝑛 1 2 𝑦 {\displaystyle{\displaystyle\int_{-1}^{1}P_{n}\left(x\right)e^{ixy}\mathrm{d}x% =i^{n}\sqrt{\frac{2\pi}{y}}J_{n+\frac{1}{2}}\left(y\right),}}
    0 references
    DLMF id
    DLMF:18.17.E19
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2abdec
    0 references
    Q10920
    Defining formula
    P n ⁡ ( x ) Legendre-spherical-polynomial 𝑛 𝑥 {\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r10.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ardec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ardec
    0 references
    Q11743
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1ndec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1rdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1pdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5760&oldid=61121"