DLMF:14.15.E28 (Q4876): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11620 / rank
 
Normal rank
Property / Symbols used: Q11620 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q11620 / qualifier
 
xml-id: C14.S15.XMD13.m1ddec
Property / Symbols used
 
Property / Symbols used: Q11621 / rank
 
Normal rank
Property / Symbols used: Q11621 / qualifier
 
Defining formula:

ζ 𝜁 {\displaystyle{\displaystyle\zeta}}

\zeta
Property / Symbols used: Q11621 / qualifier
 
xml-id: C14.S15.XMD14.m1cdec
Property / Symbols used
 
Property / Symbols used: Q11619 / rank
 
Normal rank
Property / Symbols used: Q11619 / qualifier
 
Defining formula:

α 𝛼 {\displaystyle{\displaystyle\alpha}}

\alpha
Property / Symbols used: Q11619 / qualifier
 
xml-id: C14.S15.XMD9.m1kdec

Latest revision as of 13:20, 2 January 2020

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DLMF:14.15.E28
No description defined

    Statements

    Defining formula
    1 2 ⁢ α 2 ⁢ arcsin ⁡ ( ζ α ) + 1 2 ⁢ ζ ⁢ ( α 2 - ζ 2 ) 1 / 2 = arcsin ⁡ ( x a ) - ( 1 - a 2 ) 1 / 2 ⁢ arctan ⁡ ( x ⁢ ( 1 - a 2 a 2 - x 2 ) 1 / 2 ) , 1 2 superscript 𝛼 2 𝜁 𝛼 1 2 𝜁 superscript superscript 𝛼 2 superscript 𝜁 2 1 2 𝑥 𝑎 superscript 1 superscript 𝑎 2 1 2 𝑥 superscript 1 superscript 𝑎 2 superscript 𝑎 2 superscript 𝑥 2 1 2 {\displaystyle{\displaystyle\frac{1}{2}\alpha^{2}\operatorname{arcsin}\left(% \frac{\zeta}{\alpha}\right)+\frac{1}{2}\zeta\left(\alpha^{2}-\zeta^{2}\right)^% {1/2}=\operatorname{arcsin}\left(\frac{x}{a}\right)-\left(1-a^{2}\right)^{1/2}% \operatorname{arctan}\left(x\left(\frac{1-a^{2}}{a^{2}-x^{2}}\right)^{1/2}% \right),}}
    0 references
    DLMF id
    DLMF:14.15.E28
    0 references
    constraint
    - α ≤ ζ ≤ α 𝛼 𝜁 𝛼 {\displaystyle{\displaystyle-\alpha\leq\zeta\leq\alpha}}
    0 references
    Symbols used
    Q11242
    Defining formula
    arcsin ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arcsin}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m5adec
    0 references
    Q10816
    Defining formula
    arctan ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m7aadec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1rdec
    0 references
    Q11620
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C14.S15.XMD13.m1ddec
    0 references
    Q11621
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C14.S15.XMD14.m1cdec
    0 references
    Q11619
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C14.S15.XMD9.m1kdec
    0 references
     
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