DLMF:13.21.E20 (Q4628): Difference between revisions

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Property / constraint
 

2 ⁢ κ - 2 ⁢ κ 2 - μ 2 < x ≤ 2 ⁢ κ + 2 ⁢ κ 2 - μ 2 2 𝜅 2 superscript 𝜅 2 superscript 𝜇 2 𝑥 2 𝜅 2 superscript 𝜅 2 superscript 𝜇 2 {\displaystyle{\displaystyle 2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}<x\leq 2\kappa+% 2\sqrt{\kappa^{2}-\mu^{2}}}}

2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}<x\leq 2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}
Property / constraint: 2 ⁢ κ - 2 ⁢ κ 2 - μ 2 < x ≤ 2 ⁢ κ + 2 ⁢ κ 2 - μ 2 2 𝜅 2 superscript 𝜅 2 superscript 𝜇 2 𝑥 2 𝜅 2 superscript 𝜅 2 superscript 𝜇 2 {\displaystyle{\displaystyle 2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}<x\leq 2\kappa+% 2\sqrt{\kappa^{2}-\mu^{2}}}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Q11208 / rank
 
Normal rank
Property / Symbols used: Q11208 / qualifier
 
test:

arccos ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arccos}\NVar{z}}}

\acos@@{\NVar{z}}
Property / Symbols used: Q11208 / qualifier
 
xml-id: C4.S23.SS2.p1.m6adec
Property / Symbols used
 
Property / Symbols used: Q10816 / rank
 
Normal rank
Property / Symbols used: Q10816 / qualifier
 
test:

arctan ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}

\atan@@{\NVar{z}}
Property / Symbols used: Q10816 / qualifier
 
xml-id: C4.S23.SS2.p1.m7aadec
Property / Symbols used
 
Property / Symbols used: Q11566 / rank
 
Normal rank
Property / Symbols used: Q11566 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11566 / qualifier
 
xml-id: C13.S1.XMD4.m1qdec
Property / Symbols used
 
Property / Symbols used: DLMF:13.21.E18 / rank
 
Normal rank
Property / Symbols used: DLMF:13.21.E18 / qualifier
 
test:

X 𝑋 {\displaystyle{\displaystyle X}}

X
Property / Symbols used: DLMF:13.21.E18 / qualifier
 
xml-id: C13.S21.XMD5.m1adec

Latest revision as of 15:20, 2 January 2020

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DLMF:13.21.E20
No description defined

    Statements

    test
    ζ ^ = - ( 3 2 ⁢ κ ⁢ ( - 1 2 ⁢ X + 2 ⁢ μ ⁢ arctan ⁡ ( x ⁢ κ - x ⁢ κ 2 - μ 2 - 2 ⁢ μ 2 μ ⁢ X ) + κ ⁢ arccos ⁡ ( x - 2 ⁢ κ 2 ⁢ κ 2 - μ 2 ) ) ) 2 / 3 , ^ 𝜁 superscript 3 2 𝜅 1 2 𝑋 2 𝜇 𝑥 𝜅 𝑥 superscript 𝜅 2 superscript 𝜇 2 2 superscript 𝜇 2 𝜇 𝑋 𝜅 𝑥 2 𝜅 2 superscript 𝜅 2 superscript 𝜇 2 2 3 {\displaystyle{\displaystyle\widehat{\zeta}=-\left(\frac{3}{2\kappa}\left(-% \frac{1}{2}X+2\mu\operatorname{arctan}\left(\frac{x\kappa-x\sqrt{\kappa^{2}-% \mu^{2}}-2\mu^{2}}{\mu X}\right)+\kappa\operatorname{arccos}\left(\frac{x-2% \kappa}{2\sqrt{\kappa^{2}-\mu^{2}}}\right)\right)\right)^{2/3},}}
    0 references
    DLMF id
    DLMF:13.21.E20
    0 references
    constraint
    2 ⁢ κ - 2 ⁢ κ 2 - μ 2 < x ≤ 2 ⁢ κ + 2 ⁢ κ 2 - μ 2 2 𝜅 2 superscript 𝜅 2 superscript 𝜇 2 𝑥 2 𝜅 2 superscript 𝜅 2 superscript 𝜇 2 {\displaystyle{\displaystyle 2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}<x\leq 2\kappa+% 2\sqrt{\kappa^{2}-\mu^{2}}}}
    0 references
    Symbols used
    Q11208
    test
    arccos ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arccos}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m6adec
    0 references
    Q10816
    test
    arctan ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m7aadec
    0 references
    Q11566
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1qdec
    0 references
    DLMF:13.21.E18
    test
    X 𝑋 {\displaystyle{\displaystyle X}}
    xml-id
    C13.S21.XMD5.m1adec
    0 references
     
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