L’Hôpital rule for derivatives

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31: 11.1 Special Notation

… ►Unless indicated otherwise, primes denote derivatives with respect to the argument. … ►The functions treated in this chapter are the Struve functions ${𝐇}_{\nu }\left(z\right)$ and ${𝐊}_{\nu }\left(z\right)$, the modified Struve functions ${𝐋}_{\nu }\left(z\right)$ and ${𝐌}_{\nu }\left(z\right)$, the Lommel functions $s_{\mu ,\nu }\left(z\right)$ and $S_{\mu ,\nu }\left(z\right)$, the Anger function ${𝐉}_{\nu }\left(z\right)$, the Weber function ${𝐄}_{\nu }\left(z\right)$, and the associated Anger–Weber function ${𝐀}_{\nu }\left(z\right)$.

32: 3.11 Approximation Techniques

… ►to the maximum error of the minimax polynomial $p_n(x)$ is bounded by $1+L_n$, where $L_n$ is the $n$th Lebesgue constant for Fourier series; see §1.8(i). Since $L_0=1$, $L_n$ is a monotonically increasing function of $n$, and (for example) $L_{1000}=4.07\mathrm{\dots }$, this means that in practice the gain in replacing a truncated Chebyshev-series expansion by the corresponding minimax polynomial approximation is hardly worthwhile. … ►The Padé approximants can be computed by Wynn’s cross rule. Any five approximants arranged in the Padé table as … ►By taking more derivatives into account, the smoothness of the spline will increase. …

33: 1.2 Elementary Algebra

… ►and $f^{(k)}$ is the $k$-th derivative of $f$ (§1.4(iii)). … ►This is the row times column rule. … ►

1.2.45 $${\Vert 𝐯\Vert }_p={\left(\sum _{i=1}^n{\left|v_i\right|}^p\right)}^{1/p},$$ $p\ge 1$.

ⓘ

Defines:

${\Vert 𝐯\Vert }_p$: L-p norm

Symbols:

$n$: nonnegative integer, $𝐯$: column vector and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.2.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►

1.2.48 $${\Vert 𝐯\Vert }_{\mathrm{\infty }}=\max (\left|v_1\right|,\left|v_2\right|,\mathrm{\dots },\left|v_n\right|).$$

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Defines:

${\Vert 𝐯\Vert }_{\mathrm{\infty }}$: l-infinity norm

Symbols:

$n$: nonnegative integer, $𝐯$: column vector and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.2.E48

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►

1.2.50 $$\left|⟨𝐮,𝐯⟩\right|\le {\Vert 𝐮\Vert }_p{\Vert 𝐯\Vert }_q,$$

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Symbols:

${\Vert 𝐯\Vert }_p$: L-p norm, $⟨𝐮,𝐯⟩$: inner product over vectors, $𝐯$: column vector, $𝐮$: column vector and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.2.E50

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

…

34: 18.41 Tables

… ►Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates $T_n\left(x\right)$, $U_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ for $n=0(1)12$. The ranges of $x$ are $0.2(.2)1$ for $T_n\left(x\right)$ and $U_n\left(x\right)$, and $0.5,1,3,5,10$ for $L_n\left(x\right)$ and $H_n\left(x\right)$. … ►For $P_n\left(x\right)$, $L_n\left(x\right)$, and $H_n\left(x\right)$ see §3.5(v). …

35: 19.33 Triaxial Ellipsoids

… ►The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength $H/(1+L_c\chi )$, where $L_c$ is the demagnetizing factor, given in cgs units by ►

19.33.7 $$L_c=2\pi abc{\int }_0^{\mathrm{\infty }}\frac{d\lambda }{\sqrt{(a^2+\lambda )(b^2+\lambda ){(c^2+\lambda )}^3}}=V{R}_D(a^2,b^2,c^2).$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $L_a$, $L_b$: factor and $V$: volume

Referenced by:

§19.15, §19.33(iii)

Permalink:

http://dlmf.nist.gov/19.33.E7

Encodings:

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See also:

Annotations for §19.33(iii), §19.33 and Ch.19

… ►

19.33.8 $$L_a+L_b+L_c=4\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $L_a$, $L_b$: factor

Permalink:

http://dlmf.nist.gov/19.33.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.33(iii), §19.33 and Ch.19

►where $L_a$ and $L_b$ are obtained from $L_c$ by permutation of $a$, $b$, and $c$. …

36: 23.9 Laurent and Other Power Series

… ►

23.9.1 $$c_n=(2n-1)\sum _{w\in 𝕃\setminus \{0\}}{w}^{-2n},$$ $n=2,3,4,\mathrm{\dots }$.

ⓘ

Symbols:

$\in$: element of, $\setminus$: set subtraction, $𝕃$: lattice, $n$: integer and $c_n$

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/23.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

… ►

23.9.2 $$\mathrm{\wp }\left(z\right)=\frac{1}{z^2}+\sum _{n=2}^{\mathrm{\infty }}{c}_n{z}^{2n-2},$$ ,

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $𝕃$: lattice, $n$: integer, $z$: complex and $c_n$

A&S Ref:

18.5.1

Referenced by:

§23.9, §23.9

Permalink:

http://dlmf.nist.gov/23.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

►

23.9.3 $$\zeta \left(z\right)=\frac{1}{z}-\sum _{n=2}^{\mathrm{\infty }}\frac{c_n}{2n-1}{z}^{2n-1},$$ .

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $𝕃$: lattice, $n$: integer, $z$: complex and $c_n$

A&S Ref:

18.5.5

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

… ►

23.9.6 $$\mathrm{\wp }\left({\omega }_j+t\right)=e_j+(3{e_j}^2-5c_2)t^2+(10c_2{e}_j+21c_3)t^4+(7c_2{e_j}^2+21c_3{e}_j+5c_2^2)t^6+O\left(t^8\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $c_n$ and $t$: variable

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

… ►

23.9.7 $$\sigma \left(z\right)=\sum _{m,n=0}^{\mathrm{\infty }}{a}_{m,n}{(10c_2)}^m{(56c_3)}^n\frac{z^{4m+6n+1}}{(4m+6n+1)!},$$

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Symbols:

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$): Weierstrass sigma function, $!$: factorial (as in $n!$), $𝕃$: lattice, $n$: integer, $m$: integer, $z$: complex, $c_n$ and $a_{m,n}$: coefficients

A&S Ref:

18.5.6

Referenced by:

§23.9

Permalink:

http://dlmf.nist.gov/23.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §23.9 and Ch.23

…

37: 11.15 Approximations

… ►

Luke (1975, pp. 416–421) gives Chebyshev-series expansions for ${𝐇}_n\left(x\right)$, ${𝐋}_n\left(x\right)$, $0\le \left|x\right|\le 8$, and ${𝐇}_n\left(x\right)-Y_n\left(x\right)$, $x\ge 8$, for $n=0,1$; ${\int }_0^x{t}^{-m}{𝐇}_0\left(t\right)dt$, ${\int }_0^x{t}^{-m}{𝐋}_0\left(t\right)dt$, $0\le \left|x\right|\le 8$, $m=0,1$ and ${\int }_0^x({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$, ${\int }_x^{\mathrm{\infty }}{t}^{-1}({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$, $x\ge 8$; the coefficients are to 20D.

►

MacLeod (1993) gives Chebyshev-series expansions for ${𝐋}_0\left(x\right)$, ${𝐋}_1\left(x\right)$, $0\le x\le 16$, and $I_0\left(x\right)-{𝐋}_0\left(x\right)$, $I_1\left(x\right)-{𝐋}_1\left(x\right)$, $x\ge 16$; the coefficients are to 20D.

…

38: 23.7 Quarter Periods

… ►

23.7.1 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_1\right)=e_1+\sqrt{(e_1-e_3)(e_1-e_2)}=e_1+{\omega }_1^{-2}{(K\left(k\right))}^2{k}^{\prime },$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E1

Encodings:

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See also:

Annotations for §23.7 and Ch.23

►

23.7.2 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_2\right)=e_2-\mathrm{i}\sqrt{(e_1-e_2)(e_2-e_3)}=e_2-\mathrm{i}{\omega }_1^{-2}{(K\left(k\right))}^{2}k{k}^{\prime },$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.7 and Ch.23

►

23.7.3 $$\mathrm{\wp }\left(\frac{1}{2}{\omega }_3\right)=e_3-\sqrt{(e_1-e_3)(e_2-e_3)}=e_3-{\omega }_1^{-2}{(K\left(k\right))}^{2}k,$$

ⓘ

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $e_j$: Weierstrass lattice roots, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $𝕃$: lattice, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and $k$: modulus

A&S Ref:

18.3.18

Permalink:

http://dlmf.nist.gov/23.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.7 and Ch.23

…

39: 8.19 Generalized Exponential Integral

… ►

§8.19(v) Recurrence Relation and Derivatives

… ►

$p$-Derivatives

… ►

8.19.26 $${\int }_0^{\mathrm{\infty }}{E}_p\left(t\right)E_q\left(t\right)dt=\frac{L(p)+L(q)}{p+q-1},$$ $p>0$, $q>0$, $p+q>1$,

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Symbols:

$dx$: differential of $x$ , $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $p$: parameter and $L(p)$: function

Permalink:

http://dlmf.nist.gov/8.19.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(x), §8.19 and Ch.8

►where …When $p=1,2,3,\mathrm{\dots }$, $L(p)$ can also be evaluated via (8.19.24). …

40: 1.8 Fourier Series

… ►

1.8.8 $$L_n=\frac{1}{\pi }{\int }_0^{\pi }\frac{\left|\sin \left(n+\frac{1}{2}\right)t\right|}{\sin \left(\frac{1}{2}t\right)}dt,$$ $n=0,1,\mathrm{\dots }$.

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Defines:

$L_n$: Lebesgue constants (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

… ►

1.8.9 $$L_n\sim (4/{\pi }^2)\ln n;$$

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $L_n$: Lebesgue constants

Permalink:

http://dlmf.nist.gov/1.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

… ►at every point at which $f(x)$ has both a left-hand derivative (that is, (1.4.4) applies when $h\to 0-$) and a right-hand derivative (that is, (1.4.4) applies when $h\to 0+$). … …