DLMF:1.14.E41 (Q533): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
Property / Symbols used
 
Property / Symbols used: Cauchy principal value / rank
 
Normal rank
Property / Symbols used: Cauchy principal value / qualifier
 
Defining formula:

⨍ a b superscript subscript principal-value-integral 𝑎 𝑏 {\displaystyle{\displaystyle\pvint_{\NVar{a}}^{\NVar{b}}}}

\pvint_{\NVar{a}}^{\NVar{b}}
Property / Symbols used: Cauchy principal value / qualifier
 
xml-id: C1.S4.E24.m2aadec

Latest revision as of 16:26, 1 January 2020

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DLMF:1.14.E41
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    Statements

    Defining formula
    ℋ ⁡ ( f ) ⁡ ( x ) = ℋ ⁡ f ⁡ ( x ) = 1 π ⁢ ⨍ - ∞ ∞ f ⁢ ( t ) t - x ⁢ d t , Hilbert-transform 𝑓 𝑥 Hilbert-transform 𝑓 𝑥 1 𝜋 subscript superscript principal-value-integral 𝑓 𝑡 𝑡 𝑥 𝑡 {\displaystyle{\displaystyle\mathcal{H}\left(f\right)\left(x\right)=\mathcal{H% }\mskip-3.0mu f\mskip 3.0mu \left(x\right)=\frac{1}{\pi}\pvint^{\infty}_{-% \infty}\frac{f(t)}{t-x}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:1.14.E41
    0 references
    Symbols used
    Q10895
    Defining formula
    ℋ ⁡ ( f ) ⁡ ( x ) Hilbert-transform 𝑓 𝑥 {\displaystyle{\displaystyle\mathcal{H}\left(\NVar{f}\right)\left(\NVar{x}% \right)}}
    xml-id
    C1.S14.SS5.p1.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2apdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaddec
    0 references
    Cauchy principal value
    Defining formula
    ⨍ a b superscript subscript principal-value-integral 𝑎 𝑏 {\displaystyle{\displaystyle\pvint_{\NVar{a}}^{\NVar{b}}}}
    xml-id
    C1.S4.E24.m2aadec
    0 references
     
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