sums of squares

SearchAdvancedHelp

(0.002 seconds)

41—48 of 48 matching pages

41: 4.38 Inverse Hyperbolic Functions: Further Properties

… ►

4.38.4 $$\mathrm{arccosh}z={(2(z-1))}^{1/2}\left(1+\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{1\cdot 3\cdot 5\mathrm{\cdots }(2n-1)}{2^{2n}n!(2n+1)}{(z-1)}^n\right),$$ $\mathrm{\Re }z>0$, $|z-1|\le 2$.

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Re }$: real part, $n$: integer and $z$: complex variable

Referenced by:

§4.38(i)

Permalink:

http://dlmf.nist.gov/4.38.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(i), §4.38 and Ch.4

… ►In the following equations square roots have their principal values. … ►All square roots have either possible value.

42: 14.30 Spherical and Spheroidal Harmonics

… ►

14.30.8_5 $${\mathrm{e}}^{t𝐚\cdot 𝐱}=\sqrt{4\pi }\sum _{n=0}^{\mathrm{\infty }}\sum _{m=-n}^n\frac{t^n{r}^n{\lambda }^m{Y}_{n,m}(\theta ,\varphi )}{\sqrt{(2n+1)(n+m)!(n-m)!}},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $n$: nonnegative integer and $m$: integer

Source:

Courant and Hilbert (1953, §VII.7.3)

Permalink:

http://dlmf.nist.gov/14.30.E8_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

Suggested 2019-09-26 by Anupam Garg

See also:

Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

… ►

§14.30(iii) Sums

… ►

14.30.9 $${𝖯}_l\left(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos \left({\varphi }_1-{\varphi }_2\right)\right)=\frac{4\pi }{2l+1}\sum _{m=-l}^l\overline{Y_{l,m}({\theta }_1,{\varphi }_1)}{Y}_{l,m}({\theta }_2,{\varphi }_2).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\cos z$: cosine function, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§18.18(ii)

Permalink:

http://dlmf.nist.gov/14.30.E9

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the first spherical harmonic in the sum over $m$ has been explicity marked up as a complex conjugate.

See also:

Annotations for §14.30(iii), §14.30(iii), §14.30 and Ch.14

… ►Here, in spherical coordinates, $L^2$ is the squared angular momentum operator: …

43: 19.36 Methods of Computation

… ►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. … ►The reductions in §19.29(i) represent $x,y,z$ as squares, for example $x=U_{12}^2$ in (19.29.4). … ►

19.36.13 $$2R_G(t_0^2,t_0^2+\theta c_0^2,t_0^2+\theta a_0^2)=\left(t_0^2+\theta \sum _{m=0}^{\mathrm{\infty }}{2}^{m-1}{c}_m^2\right)R_C(T^2+\theta M^2,T^2)+h_0+\sum _{m=1}^{\mathrm{\infty }}{2}^m(h_m-h_{m-1}).$$

ⓘ

Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $m$: nonnegative integer, $M$: limit, $T$: limit and $h_n$

Permalink:

http://dlmf.nist.gov/19.36.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.36(ii), §19.36(ii), §19.36 and Ch.19

…

44: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►where the branch of the square root is continuous and satisfies $\eta (\lambda )\sim \lambda -1$ as $\lambda \to 1$. … ►

8.12.7 $$S(a,\eta )\sim \frac{{\mathrm{e}}^{-\frac{1}{2}a{\eta }^2}}{\sqrt{2\pi a}}\sum _{k=0}^{\mathrm{\infty }}{c}_k(\eta )a^{-k},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $a$: parameter, $k$: nonnegative integer, $\eta (\lambda )$ , $S(a,\eta )$: function and $c_k(\eta )$: coefficients

Referenced by:

§8.12

Permalink:

http://dlmf.nist.gov/8.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

… ►

8.12.11 $$c_k(\eta )=\sum _{n=0}^{\mathrm{\infty }}{d}_{k,n}{\eta }^n,$$ ,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $k$: nonnegative integer, $n$: nonnegative integer, $\eta (\lambda )$ , $d(\pm \chi )$ and $c_k(\eta )$: coefficients

Permalink:

http://dlmf.nist.gov/8.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

… ►

8.12.13 $$\lambda -1=\eta +\frac{1}{3}{\eta }^2+\sum _{n=3}^{\mathrm{\infty }}{\alpha }_n{\eta }^n,$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer, $\eta (\lambda )$ , $\lambda$ and ${\alpha }_{n+2}$

Permalink:

http://dlmf.nist.gov/8.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

… ►

8.12.18 $$\begin{array}{c}Q(a,z)\\ P(a,z)\end{array}\}\sim \frac{z^{a-\frac{1}{2}}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a\right)}\left(d(\pm \chi )\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\chi )}{z^{k/2}}\mp \sum _{k=1}^{\mathrm{\infty }}\frac{B_k(\chi )}{z^{k/2}}\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\chi$ , $d(\pm \chi )$ , $A_k(\chi )$ and $B_k(\chi )$

Referenced by:

§8.12, Erratum (V1.0.16) for Equation (8.12.18)

Permalink:

http://dlmf.nist.gov/8.12.E18

Encodings:

pMML, png, TeX

Errata (effective with 1.0.16):

The original $\pm$ in front of the second summation was replaced by $\mp$ to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.

Reported 2017-01-28 by Richard Paris

See also:

Annotations for §8.12 and Ch.8

…

45: 4.45 Methods of Computation

… ►Then we take square roots repeatedly until $|y|$ is sufficiently small, where … ►Another method, when $x$ is large, is to sum …

46: 19.20 Special Cases

… ►

19.20.19 $$R_D(x,y,z)\sim 3x^{-1/2}{y}^{-1/2}{z}^{-1/2},$$ $z/\sqrt{xy}\to 0$.

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables and $\sim$: asymptotic equality

Referenced by:

§19.20(iv), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.20.E19

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

… ►Define $c={\sum }_{j=1}^n{b}_j$. …

47: 19.29 Reduction of General Elliptic Integrals

… ►These theorems reduce integrals over a real interval $(y,x)$ of certain integrands containing the square root of a quartic or cubic polynomial to symmetric integrals over $(0,\mathrm{\infty })$ containing the square root of a cubic polynomial (compare §19.16(i)). … ►

19.29.12 $$𝐦=(m_1,\mathrm{\dots },m_n)=\sum _{j=1}^n{m}_j{𝐞}_j,$$

ⓘ

Symbols:

$m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/19.29.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.29(ii), §19.29 and Ch.19

… ►The only cases that are integrals of the third kind are those in which at least one $m_j$ with $j>h$ is a negative integer and those in which $h=4$ and ${\sum }_{j=1}^n{m}_j$ is a positive integer. … ►

19.29.18 $$b_j^{q}I(q{𝐞}_l)=\sum _{r=0}^q\left(\genfrac{}{}{0.0pt}{}{q}{r}\right)b_l^r{d}_{lj}^{q-r}I(r{𝐞}_j),$$ $j,l=1,2,\mathrm{\dots },n$;

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $l$: nonnegative integer, $n$: nonnegative integer, $I(𝐦)$: integral and $d_{\alpha \beta }$

Referenced by:

§19.29(ii), §19.34

Permalink:

http://dlmf.nist.gov/19.29.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §19.29(ii), §19.29 and Ch.19

… ►If both square roots in (19.29.22) are 0, then the indeterminacy in the two preceding equations can be removed by using (19.27.8) to evaluate the integral as $R_G(a_1{b}_2,a_2{b}_1,0)$ multiplied either by $-2/(b_1{b}_2)$ or by $-2/(a_1{a}_2)$ in the cases of (19.29.20) or (19.29.21), respectively. …

48: 1.11 Zeros of Polynomials

… ►

… ►The sum and product of the roots are respectively $-b/a$ and $c/a$. … ►The square roots are chosen so that …