DLMF:10.15.E7 (Q3153)

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DLMF:10.15.E7
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    Statements

    Defining formula
    ∂ ⁡ J ν ⁡ ( x ) ∂ ⁡ ν | ν = - 1 2 = 2 π ⁢ x ⁢ ( Ci ⁡ ( 2 ⁢ x ) ⁢ cos ⁡ x + Si ⁡ ( 2 ⁢ x ) ⁢ sin ⁡ x ) , evaluated-at partial-derivative Bessel-J 𝜈 𝑥 𝜈 𝜈 1 2 2 𝜋 𝑥 cosine-integral 2 𝑥 𝑥 sine-integral 2 𝑥 𝑥 {\displaystyle{\displaystyle\left.\frac{\partial J_{\nu}\left(x\right)}{% \partial\nu}\right|_{\nu=-\frac{1}{2}}=\sqrt{\frac{2}{\pi x}}\left(\mathrm{Ci}% \left(2x\right)\cos x+\mathrm{Si}\left(2x\right)\sin x\right),}}
    0 references
    DLMF id
    DLMF:10.15.E7
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    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.m2afdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
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    cosine integral
    Defining formula
    Ci ⁡ ( z ) cosine-integral 𝑧 {\displaystyle{\displaystyle\mathrm{Ci}\left(\NVar{z}\right)}}
    xml-id
    C6.S2.E11.m2aadec
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    partial derivative of $$f$$ with respect to $$x$$
    Defining formula
    ∂ ⁡ f ∂ ⁡ x partial-derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
    xml-id
    C1.S5.E3.m4afdec
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    partial differential of $$x$$
    Defining formula
    ∂ ⁡ x 𝑥 {\displaystyle{\displaystyle\partial\NVar{x}}}
    xml-id
    C1.S5.E3.m2afdec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
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    sine integral
    Defining formula
    Si ⁡ ( z ) sine-integral 𝑧 {\displaystyle{\displaystyle\mathrm{Si}\left(\NVar{z}\right)}}
    xml-id
    C6.S2.E9.m2aadec
    0 references
    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1adec
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