DLMF:10.9.E30 (Q3100)

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DLMF:10.9.E30
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    Statements

    test
    J ν 2 ⁡ ( z ) + Y ν 2 ⁡ ( z ) = 8 π 2 ⁢ ∫ 0 ∞ cosh ⁡ ( 2 ⁢ ν ⁢ t ) ⁢ K 0 ⁡ ( 2 ⁢ z ⁢ sinh ⁡ t ) ⁢ d t , Bessel-J 𝜈 2 𝑧 Bessel-Y-Weber 𝜈 2 𝑧 8 superscript 𝜋 2 superscript subscript 0 2 𝜈 𝑡 modified-Bessel-second-kind 0 2 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle{J_{\nu}^{2}}\left(z\right)+{Y_{\nu}^{2}}\left(z% \right)=\frac{8}{\pi^{2}}\int_{0}^{\infty}\cosh\left(2\nu t\right)K_{0}\left(2% z\sinh t\right)\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:10.9.E30
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    constraint
    | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}}
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    Symbols used
    Bessel function of the first kind
    test
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aqdec
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    Bessel function of the second kind
    test
    Y ν ⁡ ( z ) Bessel-Y-Weber 𝜈 𝑧 {\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E3.m2agdec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aacdec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aacdec
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    hyperbolic cosine function
    test
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2ahdec
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    hyperbolic sine function
    test
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2ahdec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aacdec
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    modified Bessel function of the second kind
    test
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2adec
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    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1ahdec
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    Q11426
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1xdec
    0 references
     
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