DLMF:10.9.E6 (Q3071)

From DRMF
Revision as of 15:12, 2 January 2020 by Admin (talk | contribs) (‎Changed an Item: Add constraint)
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:10.9.E6
No description defined

    Statements

    Defining formula
    J ν ⁡ ( z ) = 1 π ⁢ ∫ 0 π cos ⁡ ( z ⁢ sin ⁡ θ - ν ⁢ θ ) ⁢ d θ - sin ⁡ ( ν ⁢ π ) π ⁢ ∫ 0 ∞ e - z ⁢ sinh ⁡ t - ν ⁢ t ⁢ d t , Bessel-J 𝜈 𝑧 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝜈 𝜃 𝜃 𝜈 𝜋 𝜋 superscript subscript 0 superscript 𝑒 𝑧 𝑡 𝜈 𝑡 𝑡 {\displaystyle{\displaystyle J_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}% \cos\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta-\frac{\sin\left(\nu\pi% \right)}{\pi}\int_{0}^{\infty}e^{-z\sinh t-\nu t}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:10.9.E6
    0 references
    constraint
    | ph ⁡ z | < 1 2 ⁢ π phase 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}}
    0 references
    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2addec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aadec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q3071&oldid=55378"