DLMF:19.27.E13 (Q6605): Difference between revisions

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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.m2akdec
Property / Symbols used
 
Property / Symbols used: Carlson’s combination of inverse circular and inverse hyperbolic functions / rank
 
Normal rank
Property / Symbols used: Carlson’s combination of inverse circular and inverse hyperbolic functions / qualifier
 
Defining formula:

R C ⁡ ( x , y ) Carlson-integral-RC 𝑥 𝑦 {\displaystyle{\displaystyle R_{C}\left(\NVar{x},\NVar{y}\right)}}

\CarlsonellintRC@{\NVar{x}}{\NVar{y}}
Property / Symbols used: Carlson’s combination of inverse circular and inverse hyperbolic functions / qualifier
 
xml-id: C19.S2.E17.m2adec
Property / Symbols used
 
Property / Symbols used: symmetric elliptic integral of third kind / rank
 
Normal rank
Property / Symbols used: symmetric elliptic integral of third kind / qualifier
 
Defining formula:

R J ⁡ ( x , y , z , p ) Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 {\displaystyle{\displaystyle R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}% \right)}}

\CarlsonsymellintRJ@{\NVar{x}}{\NVar{y}}{\NVar{z}}{\NVar{p}}
Property / Symbols used: symmetric elliptic integral of third kind / qualifier
 
xml-id: C19.S16.E2.m2abdec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
Defining formula:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2afdec
Property / Symbols used
 
Property / Symbols used: Q11888 / rank
 
Normal rank
Property / Symbols used: Q11888 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11888 / qualifier
 
xml-id: C19.S27.XMD1.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11889 / rank
 
Normal rank
Property / Symbols used: Q11889 / qualifier
 
Defining formula:

g 𝑔 {\displaystyle{\displaystyle g}}

g
Property / Symbols used: Q11889 / qualifier
 
xml-id: C19.S27.XMD5.m1ddec

Latest revision as of 14:16, 2 January 2020

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DLMF:19.27.E13
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    Statements

    Defining formula
    R J ⁡ ( x , y , z , p ) = 3 2 ⁢ z ⁢ p ⁢ ( ln ⁡ ( 8 ⁢ z a + g ) - 2 ⁢ R C ⁡ ( 1 , p z ) + O ⁡ ( ( a z + a p ) ⁢ ln ⁡ p a ) ) , Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 3 2 𝑧 𝑝 8 𝑧 𝑎 𝑔 2 Carlson-integral-RC 1 𝑝 𝑧 Big-O 𝑎 𝑧 𝑎 𝑝 𝑝 𝑎 {\displaystyle{\displaystyle R_{J}\left(x,y,z,p\right)=\frac{3}{2\sqrt{z}p}% \left(\ln\left(\frac{8z}{a+g}\right)-2R_{C}\left(1,\frac{p}{z}\right)+O\left(% \left(\frac{a}{z}+\frac{a}{p}\right)\ln\frac{p}{a}\right)\right),}}
    0 references
    DLMF id
    DLMF:19.27.E13
    0 references
    constraint
    max ⁡ ( x , y ) / min ⁡ ( z , p ) → 0 → 𝑥 𝑦 𝑧 𝑝 0 {\displaystyle{\displaystyle\max(x,y)/\min(z,p)\to 0}}
    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.m2akdec
    0 references
    Carlson’s combination of inverse circular and inverse hyperbolic functions
    Defining formula
    R C ⁡ ( x , y ) Carlson-integral-RC 𝑥 𝑦 {\displaystyle{\displaystyle R_{C}\left(\NVar{x},\NVar{y}\right)}}
    xml-id
    C19.S2.E17.m2adec
    0 references
    symmetric elliptic integral of third kind
    Defining formula
    R J ⁡ ( x , y , z , p ) Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 {\displaystyle{\displaystyle R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}% \right)}}
    xml-id
    C19.S16.E2.m2abdec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2afdec
    0 references
    Q11888
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C19.S27.XMD1.m1ddec
    0 references
    Q11889
    Defining formula
    g 𝑔 {\displaystyle{\displaystyle g}}
    xml-id
    C19.S27.XMD5.m1ddec
    0 references
     
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