DLMF:10.22.E60 (Q3434)

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DLMF:10.22.E60
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ e i ⁢ b ⁢ t ⁢ Y 0 ⁡ ( a ⁢ t ) ⁢ d t = { ( 2 ⁢ i / π ) ⁢ ( a 2 - b 2 ) - 1 2 ⁢ arcsin ⁡ ( b / a ) , 0 ≤ b < a , ( b 2 - a 2 ) - 1 2 ⁢ ( - 1 + 2 ⁢ i π ⁢ ln ⁡ ( a b + ( b 2 - a 2 ) 1 2 ) ) , 0 < a < b . superscript subscript 0 superscript 𝑒 𝑖 𝑏 𝑡 Bessel-Y-Weber 0 𝑎 𝑡 𝑡 cases 2 𝑖 𝜋 superscript superscript 𝑎 2 superscript 𝑏 2 1 2 𝑏 𝑎 0 𝑏 𝑎 superscript superscript 𝑏 2 superscript 𝑎 2 1 2 1 2 𝑖 𝜋 𝑎 𝑏 superscript superscript 𝑏 2 superscript 𝑎 2 1 2 0 𝑎 𝑏 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{ibt}Y_{0}\left(at\right)% \mathrm{d}t=\begin{cases}(2i/\pi)(a^{2}-b^{2})^{-\frac{1}{2}}\operatorname{% arcsin}\left(b/a\right),&0\leq b<a,\\ (b^{2}-a^{2})^{-\frac{1}{2}}\left(-1+\dfrac{2i}{\pi}\ln\left(\dfrac{a}{b+(b^{2% }-a^{2})^{\frac{1}{2}}}\right)\right),&0<a<b.\end{cases}}}
    0 references
    DLMF id
    DLMF:10.22.E60
    0 references
    Symbols used
    Bessel function of the second kind
    Defining formula
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2aidec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2awdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abgdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abgdec
    0 references
    Q11242
    Defining formula
    arcsin ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arcsin}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m5aadec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2abdec
    0 references
     
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