DLMF:11.10.E30 (Q4052): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11497 / rank
 
Normal rank
Property / Symbols used: Q11497 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11497 / qualifier
 
xml-id: C11.S1.XMD5.m1ddec

Latest revision as of 14:59, 2 January 2020

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DLMF:11.10.E30
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    Statements

    Defining formula
    𝐉 ν ⁡ ( z ) = 2 ⁢ sin ⁡ ( 1 2 ⁢ ν ⁢ π ) ⁢ ∑ k = 0 ∞ ( - 1 ) k ⁢ J k - 1 2 ⁢ ν + 1 2 ⁡ ( 1 2 ⁢ z ) ⁢ J k + 1 2 ⁢ ν + 1 2 ⁡ ( 1 2 ⁢ z ) + 2 ⁢ cos ⁡ ( 1 2 ⁢ ν ⁢ π ) ⁢ ∑ ′ k = 0 ′ ∞ ′ ⁢ ( - 1 ) k ⁢ J k - 1 2 ⁢ ν ⁡ ( 1 2 ⁢ z ) ⁢ J k + 1 2 ⁢ ν ⁡ ( 1 2 ⁢ z ) , Anger-J 𝜈 𝑧 2 1 2 𝜈 𝜋 superscript subscript 𝑘 0 superscript 1 𝑘 Bessel-J 𝑘 1 2 𝜈 1 2 1 2 𝑧 Bessel-J 𝑘 1 2 𝜈 1 2 1 2 𝑧 2 1 2 𝜈 𝜋 superscript subscript superscript ′ 𝑘 0 superscript 1 𝑘 Bessel-J 𝑘 1 2 𝜈 1 2 𝑧 Bessel-J 𝑘 1 2 𝜈 1 2 𝑧 {\displaystyle{\displaystyle{\mathbf{J}_{\nu}\left(z\right)=}\\ {2\sin\left(\tfrac{1}{2}\nu\pi\right)\sum_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}{% 2}\nu+\frac{1}{2}}\left(\tfrac{1}{2}z\right)J_{k+\frac{1}{2}\nu+\frac{1}{2}}% \left(\tfrac{1}{2}z\right)}+{2\cos\left(\tfrac{1}{2}\nu\pi\right)\sideset{}{{}% ^{\prime}}{\sum}_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}{2}\nu}\left(\tfrac{1}{2}z% \right)J_{k+\frac{1}{2}\nu}\left(\tfrac{1}{2}z\right)},}}
    0 references
    DLMF id
    DLMF:11.10.E30
    0 references
    Symbols used
    Anger function
    Defining formula
    𝐉 ν ⁡ ( z ) Anger-J 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{J}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C11.S10.E1.m2apdec
    0 references
    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.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.m2awdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aldec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2amdec
    0 references
    Q11491
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C11.S1.XMD2.m1aadec
    0 references
    Q11492
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C11.S1.XMD3.m1ydec
    0 references
    Q11497
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C11.S1.XMD5.m1ddec
    0 references
     
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