DLMF:13.21.E12 (Q5317): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: 13.21.E9 / rank
 
Normal rank
Property / Symbols used: 13.21.E9 / qualifier
 
DLMF defining formula:

X 𝑋 {\displaystyle{\displaystyle X}}

X
Property / Symbols used: 13.21.E9 / qualifier
 
xml-id: C13.S21.XMD2.m1bdec

Revision as of 22:29, 21 December 2021

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DLMF:13.21.E12
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    Statements

    DLMF defining formula
    κ ⁢ ζ - 4 ⁢ μ 2 - 2 ⁢ μ ⁢ arctan ⁡ ( κ ⁢ ζ - 4 ⁢ μ 2 2 ⁢ μ ) = 1 2 ⁢ ( X - π ⁢ μ ) - μ ⁢ arctan ⁡ ( x ⁢ κ - 2 ⁢ μ 2 μ ⁢ X ) + κ ⁢ arcsin ⁡ ( X 2 ⁢ κ 2 - μ 2 ) , 𝜅 𝜁 4 superscript 𝜇 2 2 𝜇 𝜅 𝜁 4 superscript 𝜇 2 2 𝜇 1 2 𝑋 𝜋 𝜇 𝜇 𝑥 𝜅 2 superscript 𝜇 2 𝜇 𝑋 𝜅 𝑋 2 superscript 𝜅 2 superscript 𝜇 2 {\displaystyle{\displaystyle\sqrt{\kappa\zeta-4\mu^{2}}-2\mu\operatorname{% arctan}\left(\frac{\sqrt{\kappa\zeta-4\mu^{2}}}{2\mu}\right)=\tfrac{1}{2}(X-% \pi\mu)-\mu\operatorname{arctan}\left(\frac{x\kappa-2\mu^{2}}{\mu X}\right)+% \kappa\operatorname{arcsin}\left(\frac{X}{2\sqrt{\kappa^{2}-\mu^{2}}}\right),}}
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    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}}\leq x<2\kappa+% 2\sqrt{\kappa^{2}-\mu^{2}}}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    Q11213
    DLMF defining formula
    arcsin ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arcsin}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m5adec
    0 references
    Q10790
    DLMF defining formula
    arctan ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m7adec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1jdec
    0 references
    13.21.E9
    DLMF defining formula
    X 𝑋 {\displaystyle{\displaystyle X}}
    xml-id
    C13.S21.XMD2.m1bdec
    0 references

    Identifiers

     
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