DLMF:10.9.E7 (Q3072): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1afdec

Revision as of 15:12, 2 January 2020

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DLMF:10.9.E7
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    Statements

    Defining formula
    Y ν ⁡ ( z ) = 1 π ⁢ ∫ 0 π sin ⁡ ( z ⁢ sin ⁡ θ - ν ⁢ θ ) ⁢ d θ - 1 π ⁢ ∫ 0 ∞ ( e ν ⁢ t + e - ν ⁢ t ⁢ cos ⁡ ( ν ⁢ π ) ) ⁢ e - z ⁢ sinh ⁡ t ⁢ d t , Bessel-Y-Weber 𝜈 𝑧 1 𝜋 superscript subscript 0 𝜋 𝑧 𝜃 𝜈 𝜃 𝜃 1 𝜋 superscript subscript 0 superscript 𝑒 𝜈 𝑡 superscript 𝑒 𝜈 𝑡 𝜈 𝜋 superscript 𝑒 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle Y_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}% \sin\left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta-\frac{1}{\pi}\int_{0}^{% \infty}\left(e^{\nu t}+e^{-\nu t}\cos\left(\nu\pi\right)\right)e^{-z\sinh t}% \mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:10.9.E7
    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 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.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aedec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
    0 references
     
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