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31—40 of 120 matching pages

31: Errata

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32: 12.11 Zeros

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12.11.5 $$p_0(\zeta )=t(\zeta ),$$

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Symbols:

$\zeta$: change of variable and $p_n(\zeta )$: coefficients

Permalink:

http://dlmf.nist.gov/12.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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12.11.6 $$p_1(\zeta )=\frac{t^3-6t}{24{(t^2-1)}^2}+\frac{5}{48{((t^2-1){\zeta }^3)}^{\frac{1}{2}}}.$$

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Symbols:

$\zeta$: change of variable and $p_n(\zeta )$: coefficients

Permalink:

http://dlmf.nist.gov/12.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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12.11.8 $$q_0(\zeta )=t(\zeta ).$$

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Symbols:

$\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/12.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

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33: 4.16 Elementary Properties

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Table 4.16.2: Trigonometric functions: quarter periods and change of sign.

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$x$	$-\theta$	$\frac{1}{2}\pi \pm \theta$	$\pi \pm \theta$	$\frac{3}{2}\pi \pm \theta$	$2\pi \pm \theta$
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34: 29.15 Fourier Series and Chebyshev Series

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29.15.1 $${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)=\frac{1}{2}{A}_0+\sum _{p=1}^n{A}_{2p}\cos \left(2p\varphi \right).$$

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Symbols:

${\mathrm{𝑢𝐸}}_{2n}^m(z,k^2)$: Lamé polynomial, $\cos z$: cosine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: change of variable and $A_{2p}$: coefficients

Referenced by:

§29.15(i), §29.15(ii)

Permalink:

http://dlmf.nist.gov/29.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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29.15.8 $${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)=\sum _{p=0}^n{A}_{2p+1}\cos \left((2p+1)\varphi \right).$$

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Symbols:

${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $\cos z$: cosine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: change of variable and $A_{2p}$: coefficients

Referenced by:

§29.15(i)

Permalink:

http://dlmf.nist.gov/29.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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29.15.13 $${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)=\sum _{p=0}^n{B}_{2p+1}\sin \left((2p+1)\varphi \right).$$

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Symbols:

${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: change of variable and $B_{2p+1}$: coefficents

Referenced by:

§29.15(i)

Permalink:

http://dlmf.nist.gov/29.15.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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29.15.18 $${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)=\mathrm{dn}(z,k)\left(\frac{1}{2}{C}_0+\sum _{p=1}^n{C}_{2p}\cos \left(2p\varphi \right)\right).$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $\cos z$: cosine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: change of variable and $C_{2p+1}$: coefficents

Referenced by:

§29.15(i)

Permalink:

http://dlmf.nist.gov/29.15.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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29.15.23 $${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)=\sum _{p=0}^n{B}_{2p+2}\sin \left((2p+2)\varphi \right).$$

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Symbols:

${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\varphi$: change of variable and $B_{2p+1}$: coefficents

Referenced by:

§29.15(i)

Permalink:

http://dlmf.nist.gov/29.15.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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35: 8.27 Approximations

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DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}{\mathrm{e}}^{x^2/2}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/2}{t}^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^2-p)+\sqrt{{(x^2-p)}^2+8(x^2+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to \mathrm{\infty }$.

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36: Philip J. Davis

… ►The surface color map can be changed from height-based to phase-based for complex valued functions, and density plots can be generated through strategic scaling. …

37: About the Project

… ►The former title of Associate Editor has been changed to Senior Associate Editor. …

38: DLMF Project News

error generating summary

39: 2.4 Contour Integrals

… ►The change of integration variable is given by ►

2.4.18 $$p(\alpha ,t)=\frac{1}{3}{w}^3+a{w}^2+bw+c,$$

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Symbols:

$p(t)$: analytic function, $w$: change of variable, $a$ , $b$ and $c$: real constant

Permalink:

http://dlmf.nist.gov/2.4.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(v), §2.4 and Ch.2

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2.4.19 $$I(\alpha ,z)={\mathrm{e}}^{-cz}{\int }_{𝒬}\exp \left(-z\left(\frac{1}{3}{w}^3+a{w}^2+bw\right)\right)f(\alpha ,w)dw,$$

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $I(\alpha ,z)$: integral, $w$: change of variable, $a$ , $b$ , $𝒬$: countour, $f(\alpha ,w)$: function and $c$: real constant

Referenced by:

§2.4(v)

Permalink:

http://dlmf.nist.gov/2.4.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(v), §2.4 and Ch.2

… ►By making a further change of variable ►

2.4.21 $$w=z^{-1/3}v-a,$$

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Symbols:

$w$: change of variable and $a$

Referenced by:

§2.4(v)

Permalink:

http://dlmf.nist.gov/2.4.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(v), §2.4 and Ch.2

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40: 2.3 Integrals of a Real Variable

… ►However, cancellation does not take place near the endpoints, owing to lack of symmetry, nor in the neighborhoods of zeros of $p^{\prime }(t)$ because $p(t)$ changes relatively slowly at these stationary points. … ►A uniform approximation can be constructed by quadratic change of integration variable: ►

2.3.25 $$p(\alpha ,t)=\frac{1}{2}{w}^2-aw+b,$$

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Symbols:

$p(\alpha ,t)$: function, $w$: change of variable, $a(\alpha )$ and $b(\alpha )$

Permalink:

http://dlmf.nist.gov/2.3.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(v), §2.3 and Ch.2

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2.3.27 $$w={(2p(\alpha ,0)-2p(\alpha ,\alpha ))}^{1/2}\pm {(2p(\alpha ,t)-2p(\alpha ,\alpha ))}^{1/2},$$

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Symbols:

$p(\alpha ,t)$: function and $w$: change of variable

Permalink:

http://dlmf.nist.gov/2.3.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(v), §2.3 and Ch.2

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2.3.31 $$f(\alpha ,w)=\sum _{s=0}^{\mathrm{\infty }}{\varphi }_s(\alpha ){(w-a)}^s,$$

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Symbols:

$w$: change of variable, $a(\alpha )$ , $f(\alpha ,w)$: function and ${\varphi }_s(\alpha )$: coefficients

Permalink:

http://dlmf.nist.gov/2.3.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(v), §2.3 and Ch.2

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