DLMF:10.17.E18 (Q3187): Difference between revisions

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Property / Symbols defined
 
Property / Symbols defined: remainder (locally) / rank
 
Normal rank
Property / constraint
 

| ph ⁡ ( z ⁢ e ∓ 1 2 ⁢ π ⁢ i ) | ≤ π phase 𝑧 superscript 𝑒 minus-or-plus 1 2 𝜋 𝑖 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(ze^{\mp\frac{1}{2}\pi i}% \right)|\leq\pi}}

|\phase@{ze^{\mp\frac{1}{2}\pi i}}|\leq\pi
Property / constraint: | ph ⁡ ( z ⁢ e ∓ 1 2 ⁢ π ⁢ i ) | ≤ π phase 𝑧 superscript 𝑒 minus-or-plus 1 2 𝜋 𝑖 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(ze^{\mp\frac{1}{2}\pi i}% \right)|\leq\pi}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: order not exceeding / rank
 
Normal rank
Property / Symbols used: order not exceeding / qualifier
 
Defining formula:

O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}

\bigO@{\NVar{x}}
Property / Symbols used: order not exceeding / qualifier
 
xml-id: C2.S1.E3.m2adec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2amdec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2afdec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2aidec
Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
Defining formula:

ph phase {\displaystyle{\displaystyle\operatorname{ph}}}

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.m1ajdec
Property / Symbols used
 
Property / Symbols used: Q11438 / rank
 
Normal rank
Property / Symbols used: Q11438 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11438 / qualifier
 
xml-id: C10.S1.XMD1.m1adec
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.m1odec
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
Defining formula:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1ndec

Latest revision as of 15:33, 2 January 2020

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DLMF:10.17.E18
No description defined

    Statements

    Defining formula
    R m , ℓ ± ⁢ ( ν , z ) = O ⁡ ( e - 2 ⁢ | z | ⁢ z - m ) , superscript subscript 𝑅 𝑚 ℓ plus-or-minus 𝜈 𝑧 Big-O superscript 𝑒 2 𝑧 superscript 𝑧 𝑚 {\displaystyle{\displaystyle R_{m,\ell}^{\pm}(\nu,z)=O\left(e^{-2|z|}z^{-m}% \right),}}
    0 references
    DLMF id
    DLMF:10.17.E18
    0 references
    constraint
    | ph ⁡ ( z ⁢ e ∓ 1 2 ⁢ π ⁢ i ) | ≤ π phase 𝑧 superscript 𝑒 minus-or-plus 1 2 𝜋 𝑖 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(ze^{\mp\frac{1}{2}\pi i}% \right)|\leq\pi}}
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.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.m2amdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aidec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1ajdec
    0 references
    Q11438
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C10.S1.XMD1.m1adec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1odec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1ndec
    0 references
     
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