DLMF:10.17.E15 (Q3184): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11426 / rank
 
Normal rank
Property / Symbols used: Q11426 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11426 / qualifier
 
xml-id: C10.S1.XMD6.m1ldec

Revision as of 15:32, 2 January 2020

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DLMF:10.17.E15
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    Statements

    Defining formula
    𝒱 z , i ⁢ ∞ ⁡ ( t - ℓ ) ≤ { | z | - ℓ , 0 ≤ ph ⁡ z ≤ π , χ ⁢ ( ℓ ) ⁢ | z | - ℓ , - 1 2 ⁢ π ≤ ph ⁡ z ≤ 0  or  π ≤ ph ⁡ z ≤ 3 2 ⁢ π , 2 ⁢ χ ⁢ ( ℓ ) ⁢ | ℑ ⁡ z | - ℓ , - π < ph ⁡ z ≤ - 1 2 ⁢ π  or  3 2 ⁢ π ≤ ph ⁡ z < 2 ⁢ π , variational-operator-on-interval 𝑧 𝑖 superscript 𝑡 ℓ cases superscript 𝑧 ℓ 0 phase 𝑧 𝜋 𝜒 ℓ superscript 𝑧 ℓ - 1 2 ⁢ π ≤ ph ⁡ z ≤ 0  or  π ≤ ph ⁡ z ≤ 3 2 ⁢ π , 2 𝜒 ℓ superscript 𝑧 ℓ - π < ph ⁡ z ≤ - 1 2 ⁢ π  or  3 2 ⁢ π ≤ ph ⁡ z < 2 ⁢ π , {\displaystyle{\displaystyle\mathcal{V}_{z,i\infty}\left(t^{-\ell}\right)\leq% \begin{cases}|z|^{-\ell},&0\leq\operatorname{ph}z\leq\pi,\\ \chi(\ell)|z|^{-\ell},&\parbox[t]{224.037pt}{$-\tfrac{1}{2}\pi\leq% \operatorname{ph}z\leq 0$ or $\pi\leq\operatorname{ph}z\leq\tfrac{3}{2}\pi$,}% \\ 2\chi(\ell)|\Im z|^{-\ell},&\parbox[t]{224.037pt}{$-\pi<\operatorname{ph}z\leq% -\tfrac{1}{2}\pi$ or $\tfrac{3}{2}\pi\leq\operatorname{ph}z<2\pi$,}\end{cases}}}
    0 references
    DLMF id
    DLMF:10.17.E15
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ajdec
    0 references
    Q10901
    Defining formula
    ℑ ⁡ absent {\displaystyle{\displaystyle\Im}}
    xml-id
    C1.S9.E2.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2agdec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1aidec
    0 references
    total variation
    Defining formula
    𝒱 a , b ⁡ ( f ) variational-operator-on-interval 𝑎 𝑏 𝑓 {\displaystyle{\displaystyle\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)}}
    xml-id
    C1.S4.E33.m2aadec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1ldec
    0 references
     
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