local maxima and minima

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1: 18.14 Inequalities

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§18.14(iii) Local Maxima and Minima

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Jacobi

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Laguerre

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2: 1.4 Calculus of One Variable

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Maxima and Minima

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3: 6.16 Mathematical Applications

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6.16.2 $$S_n(x)=\sum _{k=0}^{n-1}\frac{\sin \left((2k+1)x\right)}{2k+1}=\frac{1}{2}{\int }_0^x\frac{\sin \left(2nt\right)}{\sin t}dt=\frac{1}{2}\mathrm{Si}\left(2nx\right)+R_n(x),$$

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Defines:

$S_n(x)$: partial sum (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\mathrm{Si}\left(z\right)$: sine integral, $x$: real variable, $n$: nonnegative integer and $R_n(x)$: remainder term

Permalink:

http://dlmf.nist.gov/6.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

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6.16.3 $$R_n(x)=\frac{1}{2}{\int }_0^x\left(\frac{1}{\sin t}-\frac{1}{t}\right)\sin \left(2nt\right)dt.$$

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Defines:

$R_n(x)$: remainder term (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $x$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

… ►Hence, if $x$ is fixed and $n\to \mathrm{\infty }$, then $S_n(x)\to \frac{1}{4}\pi$, $0$, or $-\frac{1}{4}\pi$ according as , $x=0$, or ; compare (6.2.14). …

4: 1.5 Calculus of Two or More Variables

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1.5.3 $$\frac{\partial f}{\partial x}=D_{x}f=f_x=\underset{h\to 0}{lim}\frac{f(x+h,y)-f(x,y)}{h},$$

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Defines:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $D_x$: differential operator (locally)

Permalink:

http://dlmf.nist.gov/1.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §1.5(i), §1.5 and Ch.1

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§1.5(iii) Taylor’s Theorem; Maxima and Minima

… ► $f(x,y)$ has a local minimum (maximum) at $(a,b)$ if …

5: 3.11 Approximation Techniques

… ►A sufficient condition for $p_n(x)$ to be the minimax polynomial is that $\left|{ϵ}_n(x)\right|$ attains its maximum at $n+2$ distinct points in $[a,b]$ and ${ϵ}_n(x)$ changes sign at these consecutive maxima. … ►The iterative process converges locally and quadratically (§3.8(i)). … ►approximately, and the right-hand side enjoys exactly those properties concerning its maxima and minima that are required for the minimax approximation; compare Figure 18.4.3. … ►

3.11.16 $$R_{k,\mathrm{\ell }}(x)=\frac{p_0+p_{1}x+\mathrm{\cdots }+p_k{x}^k}{1+q_{1}x+\mathrm{\cdots }+q_{\mathrm{\ell }}{x}^{\mathrm{\ell }}}$$

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Defines:

$R_{k,\mathrm{\ell }}(x)$: rational approximation (locally), $p_k$: coefficients (locally) and $q_{\mathrm{\ell }}$: coefficients (locally)

Referenced by:

§3.11(iii)

Permalink:

http://dlmf.nist.gov/3.11.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(iii), §3.11 and Ch.3

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3.11.20 $$f(z)=c_0+c_{1}z+c_2{z}^2+\mathrm{\cdots }$$

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Defines:

$c_q$: coefficients (locally)

Referenced by:

§3.11(iv)

Permalink:

http://dlmf.nist.gov/3.11.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(iv), §3.11 and Ch.3

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