DLMF:10.9.E13 (Q3081)

From testwiki
Revision as of 14:13, 2 January 2020 by imported>Admin (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:10.9.E13
No description defined

    Statements

    test
    ( z + ζ z - ζ ) 1 2 ⁢ ν ⁢ J ν ⁡ ( ( z 2 - ζ 2 ) 1 2 ) = 1 π ⁢ ∫ 0 π e ζ ⁢ cos ⁡ θ ⁢ cos ⁡ ( z ⁢ sin ⁡ θ - ν ⁢ θ ) ⁢ d θ - sin ⁡ ( ν ⁢ π ) π ⁢ ∫ 0 ∞ e - ζ ⁢ cosh ⁡ t - z ⁢ sinh ⁡ t - ν ⁢ t ⁢ d t , superscript 𝑧 𝜁 𝑧 𝜁 1 2 𝜈 Bessel-J 𝜈 superscript superscript 𝑧 2 superscript 𝜁 2 1 2 1 𝜋 superscript subscript 0 𝜋 superscript 𝑒 𝜁 𝜃 𝑧 𝜃 𝜈 𝜃 𝜃 𝜈 𝜋 𝜋 superscript subscript 0 superscript 𝑒 𝜁 𝑡 𝑧 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}% \nu}J_{\nu}\left((z^{2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^% {\pi}e^{\zeta\cos\theta}\cos\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta% -\frac{\sin\left(\nu\pi\right)}{\pi}\int_{0}^{\infty}e^{-\zeta\cosh t-z\sinh t% -\nu t}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:10.9.E13
    0 references
    constraint
    ℜ ⁡ ( z + ζ ) > 0 𝑧 𝜁 0 {\displaystyle{\displaystyle\Re(z+\zeta)>0}}
    0 references
    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.m2agdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aldec
    0 references
    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aidec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aldec
    0 references
     
      Retrieved from "https://lct.wmflabs.org/w/index.php?title=Item:Q3081&oldid=16126"