DLMF:18.9.E22 (Q5618): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11207 / rank
 
Normal rank
Property / Symbols used: Q11207 / qualifier
 
Defining formula:

T n ⁡ ( x ) Chebyshev-polynomial-first-kind-T 𝑛 𝑥 {\displaystyle{\displaystyle T_{\NVar{n}}\left(\NVar{x}\right)}}

\ChebyshevpolyT{\NVar{n}}@{\NVar{x}}
Property / Symbols used: Q11207 / qualifier
 
xml-id: C18.S3.T1.t1.r4.m2afdec
Property / Symbols used
 
Property / Symbols used: Q11661 / rank
 
Normal rank
Property / Symbols used: Q11661 / qualifier
 
Defining formula:

U n ⁡ ( x ) Chebyshev-polynomial-second-kind-U 𝑛 𝑥 {\displaystyle{\displaystyle U_{\NVar{n}}\left(\NVar{x}\right)}}

\ChebyshevpolyU{\NVar{n}}@{\NVar{x}}
Property / Symbols used: Q11661 / qualifier
 
xml-id: C18.S3.T1.t1.r5.m2addec
Property / Symbols used
 
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / rank
 
Normal rank
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / qualifier
 
Defining formula:

d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}

\deriv{\NVar{f}}{\NVar{x}}
Property / Symbols used: derivative of $$f$$ with respect to $$x$$ / qualifier
 
xml-id: C1.S4.E4.m2agdec
Property / Symbols used
 
Property / Symbols used: Q11726 / rank
 
Normal rank
Property / Symbols used: Q11726 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11726 / qualifier
 
xml-id: C18.S1.XMD6.m1vdec
Property / Symbols used
 
Property / Symbols used: Q11727 / rank
 
Normal rank
Property / Symbols used: Q11727 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11727 / qualifier
 
xml-id: C18.S2.XMD3.m1udec

Latest revision as of 15:25, 2 January 2020

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DLMF:18.9.E22
No description defined

    Statements

    Defining formula
    d d x ⁡ ( ( 1 - x 2 ) 1 2 ⁢ U n ⁡ ( x ) ) = - ( n + 1 ) ⁢ ( 1 - x 2 ) - 1 2 ⁢ T n + 1 ⁡ ( x ) . derivative 𝑥 superscript 1 superscript 𝑥 2 1 2 Chebyshev-polynomial-second-kind-U 𝑛 𝑥 𝑛 1 superscript 1 superscript 𝑥 2 1 2 Chebyshev-polynomial-first-kind-T 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x^{2})^{% \frac{1}{2}}U_{n}\left(x\right)\right)=-(n+1){(1-x^{2})^{-\frac{1}{2}}}T_{n+1}% \left(x\right).}}
    0 references
    DLMF id
    DLMF:18.9.E22
    0 references
    Symbols used
    Q11207
    Defining formula
    T n ⁡ ( x ) Chebyshev-polynomial-first-kind-T 𝑛 𝑥 {\displaystyle{\displaystyle T_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r4.m2afdec
    0 references
    Q11661
    Defining formula
    U n ⁡ ( x ) Chebyshev-polynomial-second-kind-U 𝑛 𝑥 {\displaystyle{\displaystyle U_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r5.m2addec
    0 references
    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2agdec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1vdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1udec
    0 references
     
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