DLMF:10.9.E20 (Q3089): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11426 / rank
 
Normal rank
Property / Symbols used: Q11426 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11426 / qualifier
 
xml-id: C10.S1.XMD6.m1pdec

Revision as of 15:15, 2 January 2020

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DLMF:10.9.E20
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    Statements

    Defining formula
    J ν ⁡ ( z ) = Γ ⁡ ( 1 2 - ν ) ⁢ ( 1 2 ⁢ z ) ν π 3 2 ⁢ i ⁢ ∫ 0 ( 1 + ) cos ⁡ ( z ⁢ t ) ⁢ ( t 2 - 1 ) ν - 1 2 ⁢ d t , Bessel-J 𝜈 𝑧 Euler-Gamma 1 2 𝜈 superscript 1 2 𝑧 𝜈 superscript 𝜋 3 2 𝑖 superscript subscript 0 limit-from 1 𝑧 𝑡 superscript superscript 𝑡 2 1 𝜈 1 2 𝑡 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{\Gamma\left(\frac{1}{% 2}-\nu\right)(\frac{1}{2}z)^{\nu}}{\pi^{\frac{3}{2}}i}\int_{0}^{(1+)}\cos\left% (zt\right)(t^{2}-1)^{\nu-\frac{1}{2}}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:10.9.E20
    0 references
    constraint
    ν ≠ 1 2 , 3 2 , … 𝜈 1 2 3 2 … {\displaystyle{\displaystyle\nu\neq\tfrac{1}{2},\tfrac{3}{2},\ldots}}
    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.m2ajdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2asdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2akdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1asdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ahdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3asdec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1pdec
    0 references
     
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