DLMF:11.10.E31 (Q4750)

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DLMF:11.10.E31
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    Statements

    DLMF defining formula
    𝐄 ν ⁡ ( z ) = - 2 ⁢ cos ⁡ ( 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 ⁢ sin ⁡ ( 1 2 ⁢ ν ⁢ π ) ⁢ ∑ ′ k = 0 ′ ∞ ′ ⁢ ( - 1 ) k ⁢ J k - 1 2 ⁢ ν ⁡ ( 1 2 ⁢ z ) ⁢ J k + 1 2 ⁢ ν ⁡ ( 1 2 ⁢ z ) , Weber-E 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{E}_{\nu}\left(z\right)=}\\ {-2\cos\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\sin\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)},}}
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    Symbols used
    Bessel function of the first kind
    DLMF defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2addec
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    Weber function
    DLMF defining formula
    𝐄 ν ⁡ ( z ) Weber-E 𝜈 𝑧 {\displaystyle{\displaystyle\mathbf{E}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C11.S10.E2.m2ardec
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    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2axdec
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    cosine function
    DLMF defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2amdec
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    sine function
    DLMF defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2andec
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    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C11.S1.XMD2.m1abdec
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    real or complex order
    DLMF defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C11.S1.XMD3.m1zdec
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    nonnegative integer
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C11.S1.XMD5.m1edec
    0 references

    Identifiers

     
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