back substitution

SearchAdvancedHelp

(0.001 seconds)

11—20 of 65 matching pages

11: 18.40 Methods of Computation

… ►Results similar to these appear in Langhoff et al. (1976) in methods developed for physics applications, and which includes treatments of systems with discontinuities in $\mu (x)$, using what is referred to as the Stieltjes derivative which may be traced back to Stieltjes, as discussed by Deltour (1968, Eq. 12). …

12: 2.1 Definitions and Elementary Properties

… ►means that for each $n$, the difference between $f(x)$ and the $n$th partial sum on the right-hand side is $O\left({(x-c)}^n\right)$ as $x\to c$ in $𝐗$. … ►Substitution, logarithms, and powers are also permissible; compare Olver (1997b, pp. 19–22). …

13: 29.12 Definitions

… ►With the substitution $\xi ={\mathrm{sn}}^2(z,k)$ every Lamé polynomial in Table 29.12.1 can be written in the form ►

29.12.9 $${\xi }^{\rho }{(\xi -1)}^{\sigma }{(\xi -k^{-2})}^{\tau }P(\xi ),$$

ⓘ

Symbols:

$k$: real parameter, $\xi ={\mathrm{sn}}^2(z,k)$: substitution, ($0$ or $\frac{1}{2}$): parameter $\rho$ , ($0$ or $\frac{1}{2}$): parameter $\sigma$ , ($0$ or $\frac{1}{2}$): parameter $\tau$ and $P(\xi )$: polynomial

Referenced by:

§29.12(ii), §29.12(iii)

Permalink:

http://dlmf.nist.gov/29.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §29.12(ii), §29.12 and Ch.29

…

14: 8.7 Series Expansions

… ►

8.7.6 $$\mathrm{\Gamma }(a,x)=x^a{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(a)}\left(x\right)}{n+1},$$ $x>0$, .

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Proof sketch:

Integrate (18.12.13) from $z=0$ to $z=1$ and for the integral on the left-hand side use the substitution $z=\frac{t}{1+t}$. The resulting integral is (8.6.5). The constraint follows from (18.15.14).

Referenced by:

Erratum (V1.1.8) for Equation (8.7.6)

Permalink:

http://dlmf.nist.gov/8.7.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.1.8):

The constraint was updated to include .

Suggested 2022-10-14 by Walter Gautschi

See also:

Annotations for §8.7 and Ch.8

…

15: 10.13 Other Differential Equations

…

16: 19.34 Mutual Inductance of Coaxial Circles

…

17: 22.10 Maclaurin Series

… ►Further terms may be derived by substituting in the differential equations (22.13.13), (22.13.14), (22.13.15). …

18: Bibliography D

… ►

Detection of the density matrix through optical homodyne tomography without filtered back projection. Phys. Rev. A 50 (5), pp. 4298–4302.

ⓘ

External Links:

Document

Referenced by:

§13.28(iii).

Encodings:

BibTeX

…

19: 2.9 Difference Equations

… ► $c_0=1$, and higher coefficients are determined by formal substitution. … ►The coefficients $b_s$ and constant $c$ are again determined by formal substitution, beginning with $c=1$ when ${\alpha }_2-{\alpha }_1=0$, or with $b_0=1$ when ${\alpha }_2-{\alpha }_1=1,2,3,\mathrm{\dots }$. …

20: 19.28 Integrals of Elliptic Integrals

…