DLMF:14.15.E10 (Q4856): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
test:

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.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11589 / rank
 
Normal rank
Property / Symbols used: Q11589 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11589 / qualifier
 
xml-id: C14.S1.XMD1.m1gdec
Property / Symbols used
 
Property / Symbols used: Q11611 / rank
 
Normal rank
Property / Symbols used: Q11611 / qualifier
 
test:

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

\alpha
Property / Symbols used: Q11611 / qualifier
 
xml-id: C14.S15.XMD3.m1fdec
Property / Symbols used
 
Property / Symbols used: Q11614 / rank
 
Normal rank
Property / Symbols used: Q11614 / qualifier
 
test:

η 𝜂 {\displaystyle{\displaystyle\eta}}

\eta
Property / Symbols used: Q11614 / qualifier
 
xml-id: C14.S15.XMD6.m1bdec

Latest revision as of 00:49, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:14.15.E10
No description defined

    Statements

    test
    α ⁢ ln ⁡ ( ( α 2 + η 2 ) 1 / 2 + α ) - α ⁢ ln ⁡ η - ( α 2 + η 2 ) 1 / 2 = 1 2 ⁢ ln ⁡ ( ( 1 + α 2 ) ⁢ x 2 + 1 - α 2 - 2 ⁢ x ⁢ ( α 2 ⁢ x 2 - α 2 + 1 ) 1 / 2 ( x 2 - 1 ) ⁢ ( 1 - α 2 ) ) + 1 2 ⁢ α ⁢ ln ⁡ ( α 2 ⁢ ( 2 ⁢ x 2 - 1 ) + 1 + 2 ⁢ α ⁢ x ⁢ ( α 2 ⁢ x 2 - α 2 + 1 ) 1 / 2 1 - α 2 ) . 𝛼 superscript superscript 𝛼 2 superscript 𝜂 2 1 2 𝛼 𝛼 𝜂 superscript superscript 𝛼 2 superscript 𝜂 2 1 2 1 2 1 superscript 𝛼 2 superscript 𝑥 2 1 superscript 𝛼 2 2 𝑥 superscript superscript 𝛼 2 superscript 𝑥 2 superscript 𝛼 2 1 1 2 superscript 𝑥 2 1 1 superscript 𝛼 2 1 2 𝛼 superscript 𝛼 2 2 superscript 𝑥 2 1 1 2 𝛼 𝑥 superscript superscript 𝛼 2 superscript 𝑥 2 superscript 𝛼 2 1 1 2 1 superscript 𝛼 2 {\displaystyle{\displaystyle\alpha\ln\left(\left(\alpha^{2}+\eta^{2}\right)^{1% /2}+\alpha\right)-\alpha\ln\eta-\left(\alpha^{2}+\eta^{2}\right)^{1/2}=\frac{1% }{2}\ln\left(\frac{\left(1+\alpha^{2}\right)x^{2}+1-\alpha^{2}-2x\left(\alpha^% {2}x^{2}-\alpha^{2}+1\right)^{1/2}}{\left(x^{2}-1\right)\left(1-\alpha^{2}% \right)}\right)+\frac{1}{2}\alpha\ln\left(\frac{\alpha^{2}\left(2x^{2}-1\right% )+1+2\alpha x\left(\alpha^{2}x^{2}-\alpha^{2}+1\right)^{1/2}}{1-\alpha^{2}}% \right).}}
    0 references
    DLMF id
    DLMF:14.15.E10
    0 references
    Symbols used
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
    Q11589
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1gdec
    0 references
    Q11611
    test
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C14.S15.XMD3.m1fdec
    0 references
    Q11614
    test
    η 𝜂 {\displaystyle{\displaystyle\eta}}
    xml-id
    C14.S15.XMD6.m1bdec
    0 references
     
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