L’Hôpital rule for derivatives

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11: 23.6 Relations to Other Functions

… ►In this subsection $2{\omega }_1$, $2{\omega }_3$ are any pair of generators of the lattice $𝕃$, and the lattice roots $e_1$, $e_2$, $e_3$ are given by (23.3.9). … ►

23.6.13 $$\zeta \left(u\right)=\frac{{\eta }_1}{{\omega }_1}u+\frac{\pi }{2{\omega }_1}\frac{d}{dz}\ln {\theta }_1(z,q),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\ln z$: principal branch of logarithm function, $𝕃$: lattice, $q$: nome, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators, ${\eta }_j$: complex number and $u$: change of variable

Referenced by:

§23.22(i), §23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

►

23.6.14 $$\mathrm{\wp }\left(u\right)={\left(\frac{\pi }{2{\omega }_1}\right)}^2\left(\frac{{\theta }_1^{\prime \prime \prime }(0,q)}{3{\theta }_1^{\prime }(0,q)}-\frac{d^2}{{dz}^2}\ln {\theta }_1(z,q)\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\ln z$: principal branch of logarithm function, $𝕃$: lattice, $q$: nome, $z$: complex, ${\omega }_1$, ${\omega }_3$, ${\omega }_2=-{\omega }_1-{\omega }_3$: lattice generators and $u$: change of variable

Referenced by:

§23.6(i)

Permalink:

http://dlmf.nist.gov/23.6.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §23.6(i), §23.6 and Ch.23

… ►Again, in Equations (23.6.16)–(23.6.26), $2{\omega }_1,2{\omega }_3$ are any pair of generators of the lattice $𝕃$ and $e_1,e_2,e_3$ are given by (23.3.9). … ►Also, ${𝕃}_1$, ${𝕃}_2$, ${𝕃}_3$ are the lattices with generators $(4K,2\mathrm{i}{K}^{\prime })$, $(2K-2\mathrm{i}{K}^{\prime },2K+2\mathrm{i}{K}^{\prime })$, $(2K,4\mathrm{i}{K}^{\prime })$, respectively. …

12: 18.39 Applications in the Physical Sciences

… ►where ${\mathrm{L}}^2$ is the (squared) angular momentum operator (14.30.12). … ►with an infinite set of orthonormal $L^2$ eigenfunctions …$p$ here being the order of the Laguerre polynomial, $L_p^{(2l+1)}$ of Table 18.8.1, line 11, and $l$ the angular momentum quantum number, and where … ►The bound state $L^2$ eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the $\delta$-function normalized (non-$L^2$) in Chapter 33, where the solutions appear as Whittaker functions. … ►The fact that non-$L^2$ continuum scattering eigenstates may be expressed in terms or (infinite) sums of $L^2$ functions allows a reformulation of scattering theory in atomic physics wherein no non-$L^2$ functions need appear. …

13: 29.1 Special Notation

… ►All derivatives are denoted by differentials, not by primes. … ►The relation to the Lamé functions $L_{c\nu }^{(m)}$, $L_{s\nu }^{(m)}$of Jansen (1977) is given by ►

$${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)={(-1)}^m{L}_{c\nu }^{(2m)}(\psi ,k_{}^{\prime }{}_{}{}^2),$$

►

$${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)={(-1)}^m{L}_{s\nu }^{(2m+1)}(\psi ,k_{}^{\prime }{}_{}{}^2),$$

►

$${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)={(-1)}^m{L}_{c\nu }^{(2m+1)}(\psi ,k_{}^{\prime }{}_{}{}^2),$$

…

14: 18.5 Explicit Representations

… ►

18.5.6 $$L_n^{(\alpha )}\left(\frac{1}{x}\right)=\frac{{(-1)}^n}{n!}{x}^{n+\alpha +1}{\mathrm{e}}^{1/x}\frac{d^n}{{dx}^n}\left(x^{-\alpha -1}{\mathrm{e}}^{-1/x}\right).$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.5(ii)

Permalink:

http://dlmf.nist.gov/18.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(ii), §18.5 and Ch.18

… ►Similarly in the cases of the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$ and the Laguerre polynomials $L_n^{(\alpha )}\left(x\right)$ we assume that $\lambda >-\frac{1}{2},\lambda \ne 0$, and $\alpha >-1$, unless stated otherwise. … ►

$L_0\left(x\right)=1,$

… ►

$L_0^{(\alpha )}\left(x\right)=1,$

►

$L_1^{(\alpha )}\left(x\right)=-x+\alpha +1,$

…

15: 18.17 Integrals

… ►

18.17.2 $${\int }_0^x{L}_m\left(y\right)L_n\left(x-y\right)dy={\int }_0^x{L}_{m+n}\left(y\right)dy=L_{m+n}\left(x\right)-L_{m+n+1}\left(x\right).$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $L_n\left(x\right)=L_n^{(0)}\left(x\right)$: Laguerre polynomial, $y$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(i)

Permalink:

http://dlmf.nist.gov/18.17.E2

Encodings:

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See also:

Annotations for §18.17(i), §18.17(i), §18.17 and Ch.18

… ►Formulas (18.17.9), (18.17.10) and (18.17.11) are fractional generalizations of $n$-th derivative formulas which are, after substitution of (18.5.7), special cases of (15.5.4), (15.5.5) and (15.5.3), respectively. … ►

18.17.15 $${\mathrm{e}}^{-x}{L}_n^{(\alpha )}\left(x\right)={\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-y}{L}_n^{(\alpha +\mu )}\left(y\right)\frac{{(y-x)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$ $\mu >0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(iv), §18.17(iv)

Permalink:

http://dlmf.nist.gov/18.17.E15

Encodings:

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See also:

Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

►Formulas (18.17.14) and (18.17.15) are fractional generalizations of $n$-th derivative formulas which are, after substitution of (13.6.19), special cases of (13.3.18) and (13.3.20), respectively. … ►

18.17.47 $${\int }_0^x{t}^{\alpha }\frac{L_m^{(\alpha )}\left(t\right)}{L_m^{(\alpha )}\left(0\right)}{(x-t)}^{\beta }\frac{L_n^{(\beta )}\left(x-t\right)}{L_n^{(\beta )}\left(0\right)}dt=\frac{\mathrm{\Gamma }\left(\alpha +1\right)\mathrm{\Gamma }\left(\beta +1\right)}{\mathrm{\Gamma }\left(\alpha +\beta +2\right)}{x}^{\alpha +\beta +1}\frac{L_{m+n}^{(\alpha +\beta +1)}\left(x\right)}{L_{m+n}^{(\alpha +\beta +1)}\left(0\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$ , $\int$: integral, $t$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(viii)

Permalink:

http://dlmf.nist.gov/18.17.E47

Encodings:

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See also:

Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

…

16: 11.2 Definitions

… ►The functions $z^{-\nu -1}{𝐇}_{\nu }\left(z\right)$ and $z^{-\nu -1}{𝐋}_{\nu }\left(z\right)$ are entire functions of $z$ and $\nu$. … ►

11.2.4 $${𝐋}_0\left(z\right)=\frac{2}{\pi }\left(z+\frac{z^3}{1^2\cdot 3^2}+\frac{z^5}{1^2\cdot 3^2\cdot 5^2}+\mathrm{\cdots }\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${𝐋}_{\nu }\left(z\right)$: modified Struve function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/11.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(i), §11.2 and Ch.11

… ►Unless indicated otherwise, ${𝐇}_{\nu }\left(z\right)$, ${𝐊}_{\nu }\left(z\right)$, ${𝐋}_{\nu }\left(z\right)$, and ${𝐌}_{\nu }\left(z\right)$ assume their principal values throughout the DLMF. … ►

11.2.10 $$w={𝐋}_{\nu }\left(z\right),{𝐌}_{\nu }\left(z\right).$$

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Symbols:

${𝐌}_{\nu }\left(z\right)$: modified Struve function, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(ii), §11.2(ii), §11.2 and Ch.11

… ►

11.2.16 $$w={𝐋}_{\nu }\left(z\right)+A{K}_{\nu }\left(z\right)+B{I}_{\nu }\left(z\right),$$

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Referenced by:

§11.2(iii)

Permalink:

http://dlmf.nist.gov/11.2.E16

Encodings:

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See also:

Annotations for §11.2(iii), §11.2 and Ch.11

…

17: 18.4 Graphics

… ►

►►

Figure 18.4.5: Laguerre polynomials $L_n\left(x\right)$, $n=1,2,3,4,5$. Magnify

►

►►

Figure 18.4.6: Laguerre polynomials $L_3^{(\alpha )}\left(x\right)$, $\alpha =0,1,2,3,4$. Magnify

… ►

►

►

Figure 18.4.8: Laguerre polynomials $L_3^{(\alpha )}\left(x\right)$, $0\le \alpha \le 3$, $0\le x\le 10$. Magnify 3D Help

►

►

►

Figure 18.4.9: Laguerre polynomials $L_4^{(\alpha )}\left(x\right)$, $0\le \alpha \le 3$, $0\le x\le 10$. Magnify 3D Help

18: 3.2 Linear Algebra

… ►With $𝐲={[y_1,y_2,\mathrm{\dots },y_n]}^{\mathrm{T}}$ the process of solution can then be regarded as first solving the equation $𝐋𝐲=𝐛$ for $𝐲$ (forward elimination), followed by the solution of $𝐔𝐱=𝐲$ for $𝐱$ (back substitution). … ►Because of rounding errors, the residual vector $𝐫=𝐛-𝐀𝐱$ is nonzero as a rule. … ►

3.2.8

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Defines:

${\mathrm{\ell }}_{jk}$: elements of $𝐋$ (locally)

Permalink:

http://dlmf.nist.gov/3.2.E8

Encodings:

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See also:

Annotations for §3.2(ii), §3.2 and Ch.3

… ►In the case that the orthogonality condition is replaced by $𝐒$-orthogonality, that is, ${𝐯}_j^{\mathrm{T}}𝐒{𝐯}_k={\delta }_{j,k}$, $j,k=1,2,\mathrm{\dots },n$, for some positive definite matrix $𝐒$ with Cholesky decomposition $𝐒={𝐋}^{\mathrm{T}}𝐋$, then the details change as follows. … ►

${𝐯}_{j+1}={𝐋}^{-1}{\left({𝐋}^{-1}\right)}^{\mathrm{T}}𝐮/{\beta }_{j+1},$

…

19: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

§1.18(ii) $L^2$ spaces on intervals in $ℝ$

… ►where the limit has to be understood in the sense of $L^2$ convergence in the mean: … ►Eigenfunctions corresponding to the continuous spectrum are non-$L^2$ functions. … ►The well must be deep and broad enough to allow existence of such $L^2$ discrete states. … ►, $f\in C^2(X)$) of $Lf=zf$ which is moreover in $L^2\left(X\right)$. …

20: 31.2 Differential Equations

… ►

31.2.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\gamma }{z}+\frac{\delta }{z-1}+\frac{ϵ}{z-a}\right)\frac{dw}{dz}+\frac{\alpha \beta z-q}{z(z-1)(z-a)}w=0,$$ $\alpha +\beta +1=\gamma +\delta +ϵ$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§29.2(ii), §31.11(iii), §31.12, §31.13, §31.14(i), §31.17(i), §31.18, §31.2(v), §31.2(v), §31.2(v), §31.2(i), §31.3(i), §31.3(i), §31.3(ii), §31.3(ii), §31.3(ii), §31.3(iii), §31.3(iii), §31.4, §31.5, §31.6, §31.7(ii), §31.8, §31.8, §31.8

Permalink:

http://dlmf.nist.gov/31.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(i), §31.2 and Ch.31

… ►

31.2.6 $$\frac{d^{2}w}{{d\theta }^2}+\left((2\gamma -1)\cot \theta -(2\delta -1)\tan \theta -\frac{ϵ\sin \left(2\theta \right)}{a-{\sin }^2\theta }\right)\frac{dw}{d\theta }+4\frac{\alpha \beta {\sin }^2\theta -q}{a-{\sin }^2\theta }w=0.$$

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Symbols:

$\cot z$: cotangent function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function, $\tan z$: tangent function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.2.E6

Encodings:

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See also:

Annotations for §31.2(iii), §31.2 and Ch.31

… ►where $2{\omega }_1$ and $2{\omega }_3$ with $\mathrm{\Im }\left({\omega }_3/{\omega }_1\right)>0$ are generators of the lattice $𝕃$ for $\mathrm{\wp }\left(z|𝕃\right)$. … ►

31.2.10 $$w(\xi )={\left(\mathrm{\wp }\left(\xi \right)-e_3\right)}^{(1-2\gamma )/4}{\left(\mathrm{\wp }\left(\xi \right)-e_2\right)}^{(1-2\delta )/4}{\left(\mathrm{\wp }\left(\xi \right)-e_1\right)}^{(1-2ϵ)/4}W(\xi ),$$

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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, $e_j$: Weierstrass lattice roots, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $𝕃$: lattice and $W(\xi )$: solution

Permalink:

http://dlmf.nist.gov/31.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

… ►

31.2.11 $$d^{2}W/{d\xi }^2+\left(H+b_0\mathrm{\wp }\left(\xi \right)+b_1\mathrm{\wp }\left(\xi +{\omega }_1\right)+b_2\mathrm{\wp }\left(\xi +{\omega }_2\right)+b_3\mathrm{\wp }\left(\xi +{\omega }_3\right)\right)W=0,$$

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Defines:

$W(\xi )$: solution (locally)

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${\omega }_j$: nonzero complex number, $𝕃$: lattice, $b_j$: coefficients and $H$

Permalink:

http://dlmf.nist.gov/31.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

…