DLMF:10.22.E61 (Q4132)

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DLMF:10.22.E61
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    Statements

    DLMF defining formula
    ∫ 0 ∞ t - 1 ⁢ e i ⁢ b ⁢ t ⁢ J μ ⁡ ( a ⁢ t ) ⁢ d t = { ( 1 / μ ) ⁢ exp ⁡ ( i ⁢ μ ⁢ arcsin ⁡ ( b / a ) ) , 0 ≤ b ≤ a , a μ ⁢ exp ⁡ ( 1 2 ⁢ μ ⁢ π ⁢ i ) μ ⁢ ( b + ( b 2 - a 2 ) 1 2 ) μ , 0 < a ≤ b . superscript subscript 0 superscript 𝑡 1 superscript 𝑒 𝑖 𝑏 𝑡 Bessel-J 𝜇 𝑎 𝑡 𝑡 cases 1 𝜇 𝑖 𝜇 𝑏 𝑎 0 𝑏 𝑎 superscript 𝑎 𝜇 1 2 𝜇 𝜋 𝑖 𝜇 superscript 𝑏 superscript superscript 𝑏 2 superscript 𝑎 2 1 2 𝜇 0 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}t^{-1}e^{ibt}J_{\mu}\left(at% \right)\mathrm{d}t=\begin{cases}(1/\mu)\exp\left(i\mu\operatorname{arcsin}% \left(b/a\right)\right),&0\leq b\leq a,\\ \dfrac{a^{\mu}\exp\left(\frac{1}{2}\mu\pi i\right)}{\mu\left(b+(b^{2}-a^{2})^{% \frac{1}{2}}\right)^{\mu}},&0<a\leq b.\end{cases}}}
    0 references
    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.m2aatdec
    0 references
    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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    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abhdec
    0 references
    exponential function
    DLMF defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2aedec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abhdec
    0 references
    Q11213
    DLMF defining formula
    arcsin ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arcsin}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m5abdec
    0 references

    Identifiers

     
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