DLMF:10.22.E59 (Q4130)

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DLMF:10.22.E59
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    Statements

    DLMF defining formula
    ∫ 0 ∞ e i ⁢ b ⁢ t ⁢ J μ ⁡ ( a ⁢ t ) ⁢ d t = { exp ⁡ ( i ⁢ μ ⁢ arcsin ⁡ ( b / a ) ) ( a 2 - b 2 ) 1 2 , 0 ≤ b < a , i ⁢ a μ ⁢ exp ⁡ ( 1 2 ⁢ μ ⁢ π ⁢ i ) ( b 2 - a 2 ) 1 2 ⁢ ( b + ( b 2 - a 2 ) 1 2 ) μ , 0 < a < b . superscript subscript 0 superscript 𝑒 𝑖 𝑏 𝑡 Bessel-J 𝜇 𝑎 𝑡 𝑡 cases 𝑖 𝜇 𝑏 𝑎 superscript superscript 𝑎 2 superscript 𝑏 2 1 2 0 𝑏 𝑎 𝑖 superscript 𝑎 𝜇 1 2 𝜇 𝜋 𝑖 superscript superscript 𝑏 2 superscript 𝑎 2 1 2 superscript 𝑏 superscript superscript 𝑏 2 superscript 𝑎 2 1 2 𝜇 0 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{ibt}J_{\mu}\left(at\right)% \mathrm{d}t=\begin{cases}\dfrac{\exp\left(i\mu\operatorname{arcsin}\left(b/a% \right)\right)}{(a^{2}-b^{2})^{\frac{1}{2}}},&0\leq b<a,\\ \dfrac{ia^{\mu}\exp\left(\frac{1}{2}\mu\pi i\right)}{(b^{2}-a^{2})^{\frac{1}{2% }}\left(b+(b^{2}-a^{2})^{\frac{1}{2}}\right)^{\mu}},&0<a<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.m2aasdec
    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.m2avdec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abfdec
    0 references
    exponential function
    DLMF defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2addec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abfdec
    0 references
    Q11213
    DLMF defining formula
    arcsin ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arcsin}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m5adec
    0 references

    Identifiers

     
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