combined

SearchAdvancedHelp

(0.000 seconds)

11—20 of 163 matching pages

11: 10.33 Continued Fractions

…

12: 13.25 Products

…

13: 27.15 Chinese Remainder Theorem

… ►These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

14: 28.34 Methods of Computation

… ►

Solution of (28.2.1) by boundary-value methods; see §3.7(iii). This can be combined with §28.34(ii)(c).

►

Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

…

15: 9.8 Modulus and Phase

… ►

9.8.9 $${|x|}^{1/2}{M}^2\left(x\right)=\frac{1}{2}\xi \left(J_{1/3}^2\left(\xi \right)+Y_{1/3}^2\left(\xi \right)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $M\left(z\right)$: Airy modulus function, $x$: real variable and $\xi$: change of variable

Proof sketch:

Combine definition (9.8.3) with (9.6.17) and (9.6.19) using (10.4.3).

Referenced by:

(9.8.20), (9.8.21), (9.8.22), (9.8.23)

Permalink:

http://dlmf.nist.gov/9.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

►

9.8.10 $${|x|}^{-1/2}{N}^2\left(x\right)=\frac{1}{2}\xi \left(J_{2/3}^2\left(\xi \right)+Y_{2/3}^2\left(\xi \right)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $N\left(z\right)$: Airy modulus function, $x$: real variable and $\xi$: change of variable

Proof sketch:

Combine definition (9.8.7) with (9.6.18) and (9.6.20) using (10.4.3).

Permalink:

http://dlmf.nist.gov/9.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

►

9.8.11 $$\theta \left(x\right)=\frac{2}{3}\pi +\arctan \left(Y_{1/3}\left(\xi \right)/J_{1/3}\left(\xi \right)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\theta \left(z\right)$: Airy phase function, $x$: real variable and $\xi$: change of variable

Proof sketch:

Combine definition (9.8.4) with (9.6.17) and (9.6.19) using (10.4.3).

Permalink:

http://dlmf.nist.gov/9.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

►

9.8.12 $$\varphi \left(x\right)=\frac{1}{3}\pi +\arctan \left(Y_{2/3}\left(\xi \right)/J_{2/3}\left(\xi \right)\right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\varphi \left(z\right)$: Airy phase function, $x$: real variable and $\xi$: change of variable

Proof sketch:

Combine definition (9.8.8) with (9.6.18) and (9.6.20) using (10.4.3).

Referenced by:

(9.8.20), (9.8.21), (9.8.22), (9.8.23)

Permalink:

http://dlmf.nist.gov/9.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(i), §9.8 and Ch.9

… ►

9.8.19 $$\theta _{}^{\prime }{}_{}{}^2\left(x\right)+\frac{1}{2}({\theta }^{\prime \prime \prime }\left(x\right)/{\theta }^{\prime }\left(x\right))-\frac{3}{4}{({\theta }^{\prime \prime }\left(x\right)/{\theta }^{\prime }\left(x\right))}^2=-x.$$

ⓘ

Symbols:

$\theta \left(z\right)$: Airy phase function and $x$: real variable

Proof sketch:

Combine (9.8.4) with (9.2.1).

Permalink:

http://dlmf.nist.gov/9.8.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(ii), §9.8 and Ch.9

…

16: 19.17 Graphics

…

17: 10.44 Sums

…

18: 19.12 Asymptotic Approximations

… ►

19.12.4 $$(1-{\alpha }^2)\mathrm{\Pi }({\alpha }^2,k)=\left(\ln \frac{4}{k^{\prime }}\right)\left(1+O\left(k_{}^{\prime }{}_{}{}^2\right)\right)-{\alpha }^2{R}_C(1,1-{\alpha }^2),$$ ,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\ln z$: principal branch of logarithm function, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

►

19.12.5 $$(1-{\alpha }^2)\mathrm{\Pi }({\alpha }^2,k)=\left(\ln \left(\frac{4}{k^{\prime }}\right)-R_C(1,1-{\alpha }^{-2})\right)\left(1+O\left(k_{}^{\prime }{}_{}{}^2\right)\right),$$ .

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $\ln z$: principal branch of logarithm function, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

… ►

19.12.6 $$R_C(x,y)=\frac{\pi }{2\sqrt{y}}-\frac{\sqrt{x}}{y}\left(1+O\left(\sqrt{\frac{x}{y}}\right)\right),$$ $x/y\to 0$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\pi$: the ratio of the circumference of a circle to its diameter

Referenced by:

§19.12

Permalink:

http://dlmf.nist.gov/19.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

►

19.12.7 $$R_C(x,y)=\frac{1}{2\sqrt{x}}\left(\left(1+\frac{y}{2x}\right)\ln \left(\frac{4x}{y}\right)-\frac{y}{2x}\right)(1+O\left(y^2/x^2\right)),$$ $y/x\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\ln z$: principal branch of logarithm function

Referenced by:

§19.12, §19.2(iv)

Permalink:

http://dlmf.nist.gov/19.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.12 and Ch.19

19: 10.51 Recurrence Relations and Derivatives

…

20: 13.11 Series

… ►

13.11.2 $$M(a,b,z)=\mathrm{\Gamma }\left(b-a-\frac{1}{2}\right){\mathrm{e}}^{\frac{1}{2}z}{\left(\frac{1}{4}z\right)}^{a-b+\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(2b-2a-1\right)}_s{\left(b-2a\right)}_s(b-a-\frac{1}{2}+s)}{{\left(b\right)}_{s}s!}{I}_{b-a-\frac{1}{2}+s}\left(\frac{1}{2}z\right),$$ $b-a+\frac{1}{2},b\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $s$: nonnegative integer and $z$: complex variable

Proof sketch:

Combine (13.24.1) with (13.14.4).

A&S Ref:

13.3.6

Referenced by:

§13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.11.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.11 and Ch.13

… ►For additional expansions combine (13.14.4), (13.14.5), and §13.24. …