DLMF:13.7.E13 (Q4425): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11557 / rank
 
Normal rank
Property / Symbols used: Q11557 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11557 / qualifier
 
xml-id: C13.S1.XMD6.m1jdec
Property / Symbols used
 
Property / Symbols used: Q11560 / rank
 
Normal rank
Property / Symbols used: Q11560 / qualifier
 
Defining formula:

δ 𝛿 {\displaystyle{\displaystyle\delta}}

\delta
Property / Symbols used: Q11560 / qualifier
 
xml-id: C13.S1.XMD7.m1bdec
Property / Symbols used
 
Property / Symbols used: remainder (locally) / rank
 
Normal rank
Property / Symbols used: remainder (locally) / qualifier
 
Defining formula:

R n ⁢ ( a , b , z ) subscript 𝑅 𝑛 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle R_{n}(a,b,z)}}

R_{n}(a,b,z)
Property / Symbols used: remainder (locally) / qualifier
 
xml-id: C13.S7.XMD9.m1bdec

Latest revision as of 15:56, 2 January 2020

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English
DLMF:13.7.E13
No description defined

    Statements

    Defining formula
    R m , n ⁢ ( a , b , z ) = { O ⁡ ( e - | z | ⁢ z - m ) , | ph ⁡ z | ≤ π , O ⁡ ( e z ⁢ z - m ) , π ≤ | ph ⁡ z | ≤ 5 2 ⁢ π - δ . subscript 𝑅 𝑚 𝑛 𝑎 𝑏 𝑧 cases Big-O superscript 𝑒 𝑧 superscript 𝑧 𝑚 phase 𝑧 𝜋 Big-O superscript 𝑒 𝑧 superscript 𝑧 𝑚 𝜋 phase 𝑧 5 2 𝜋 𝛿 {\displaystyle{\displaystyle R_{m,n}(a,b,z)=\begin{cases}O\left(e^{-|z|}z^{-m}% \right),&|\operatorname{ph}z|\leq\pi,\\ O\left(e^{z}z^{-m}\right),&\pi\leq|\operatorname{ph}z|\leq\tfrac{5}{2}\pi-% \delta.\\ \end{cases}}}
    0 references
    DLMF id
    DLMF:13.7.E13
    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.m2aedec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1abdec
    0 references
    Q11561
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C13.S1.XMD1.m1adec
    0 references
    Q11559
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C13.S1.XMD2.m1fdec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1jdec
    0 references
    Q11560
    Defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C13.S1.XMD7.m1bdec
    0 references
    remainder (locally)
    Defining formula
    R n ⁢ ( a , b , z ) subscript 𝑅 𝑛 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle R_{n}(a,b,z)}}
    xml-id
    C13.S7.XMD9.m1bdec
    0 references
     
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