DLMF:10.23.E17 (Q3471): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11426 / rank
Normal rank
 
Property / Symbols used: Q11426 / qualifier
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
 
Property / Symbols used: Q11426 / qualifier
xml-id: C10.S1.XMD6.m1jdec
 

Latest revision as of 13:32, 2 January 2020

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DLMF:10.23.E17
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    Statements

    test
    Y n ⁡ ( z ) = - n ! ⁢ ( 1 2 ⁢ z ) - n π ⁢ ∑ k = 0 n - 1 ( 1 2 ⁢ z ) k ⁢ J k ⁡ ( z ) k ! ⁢ ( n - k ) + 2 π ⁢ ( ln ⁡ ( 1 2 ⁢ z ) - ψ ⁡ ( n + 1 ) ) ⁢ J n ⁡ ( z ) - 2 π ⁢ ∑ k = 1 ∞ ( - 1 ) k ⁢ ( n + 2 ⁢ k ) ⁢ J n + 2 ⁢ k ⁡ ( z ) k ⁢ ( n + k ) , Bessel-Y-Weber 𝑛 𝑧 𝑛 superscript 1 2 𝑧 𝑛 𝜋 superscript subscript 𝑘 0 𝑛 1 superscript 1 2 𝑧 𝑘 Bessel-J 𝑘 𝑧 𝑘 𝑛 𝑘 2 𝜋 1 2 𝑧 digamma 𝑛 1 Bessel-J 𝑛 𝑧 2 𝜋 superscript subscript 𝑘 1 superscript 1 𝑘 𝑛 2 𝑘 Bessel-J 𝑛 2 𝑘 𝑧 𝑘 𝑛 𝑘 {\displaystyle{\displaystyle Y_{n}\left(z\right)=-\frac{n!(\tfrac{1}{2}z)^{-n}% }{\pi}\sum_{k=0}^{n-1}\frac{(\tfrac{1}{2}z)^{k}J_{k}\left(z\right)}{k!(n-k)}+% \frac{2}{\pi}\left(\ln\left(\tfrac{1}{2}z\right)-\psi\left(n+1\right)\right)J_% {n}\left(z\right)-\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{(n+2k)J_{n+2k}% \left(z\right)}{k(n+k)},}}
    0 references
    DLMF id
    DLMF:10.23.E17
    0 references
    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aldec
    0 references
    Bessel function of the second kind
    test
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2adec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
    Q11429
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C10.S1.XMD2.m1ddec
    0 references
    Q11428
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1odec
    0 references
     
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