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41: 36.12 Uniform Approximation of Integrals

… ►The function $g$ has a smooth amplitude. … ►

36.12.4 $$f(u(t,𝐲);𝐲)=A(𝐲)+{\mathrm{\Phi }}_K(t;𝐱(𝐲)),$$

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Defines:

$u(t;𝐲)$: mapping (locally)

Symbols:

${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $t$: variable, $K$: codimension, $f(u,𝐲)$: function, $𝐱(𝐲)$: function and $A(𝐲)$: function

Permalink:

http://dlmf.nist.gov/36.12.E4

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Annotations for §36.12(i), §36.12 and Ch.36

►with the $K+1$ functions $A(𝐲)$ and $𝐱(𝐲)$ determined by correspondence of the $K+1$ critical points of $f$ and ${\mathrm{\Phi }}_K$. … ►

36.12.5 $$f(u_j(𝐲);𝐲)=A(𝐲)+{\mathrm{\Phi }}_K(t_j(𝐱(𝐲));𝐱(𝐲)),$$

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Symbols:

${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $K$: codimension, $f(u,𝐲)$: function, $u(t;𝐲)$: mapping, $𝐱(𝐲)$: function, $A(𝐲)$: function and $t_j(𝐱)$: solutions

Permalink:

http://dlmf.nist.gov/36.12.E5

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Annotations for §36.12(i), §36.12 and Ch.36

… ►

36.12.6 $$A(𝐲)=f(u(0,𝐲);𝐲),$$

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Defines:

$A(𝐲)$: function (locally)

Symbols:

$f(u,𝐲)$: function and $u(t;𝐲)$: mapping

Permalink:

http://dlmf.nist.gov/36.12.E6

Encodings:

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See also:

Annotations for §36.12(i), §36.12 and Ch.36

…

42: 13.2 Definitions and Basic Properties

… ►

13.2.4 $$M(a,b,z)=\mathrm{\Gamma }\left(b\right)𝐌(a,b,z).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.2(ii)

Permalink:

http://dlmf.nist.gov/13.2.E4

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Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

… ► $M(a,b,z)$ is entire in $z$ and $a$, and is a meromorphic function of $b$. … ►

13.2.5 $$\underset{b\to -n}{lim}\frac{M(a,b,z)}{\mathrm{\Gamma }\left(b\right)}=𝐌(a,-n,z)=\frac{{\left(a\right)}_{n+1}}{(n+1)!}{z}^{n+1}M(a+n+1,n+2,z).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E5

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See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

… ►

13.2.14 $$U(-n,b,z)={(-1)}^n{\left(b\right)}_n+O\left(z\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E14

Encodings:

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Annotations for §13.2(iii), §13.2 and Ch.13

… ►

13.2.35 $$𝒲\left\{𝐌(a,b,z),{\mathrm{e}}^{z}U(b-a,b,{\mathrm{e}}^{\pm \pi \mathrm{i}}z)\right\}={\mathrm{e}}^{\mp b\pi \mathrm{i}}{z}^{-b}{\mathrm{e}}^z/\mathrm{\Gamma }\left(b-a\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

13.1.23

Permalink:

http://dlmf.nist.gov/13.2.E35

Encodings:

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See also:

Annotations for §13.2(vi), §13.2 and Ch.13

…

43: 12.20 Approximations

… ►Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions $U(a,b,x)$ and $M(a,b,x)$ (§13.2(i)) whose regions of validity include intervals with endpoints $x=\mathrm{\infty }$ and $x=0$, respectively. …

44: Bibliography L

… ►

Monotonicity properties of zeros of generalized Airy functions. Z. Angew. Math. Phys. 39 (2), pp. 267–271.

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External Links:

ISSN 0044-2275, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§9.13(i).

Encodings:

BibTeX

… ►

Further inequalities for the gamma function. Math. Comp. 42 (166), pp. 597–600.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

BibTeX

►

Inequalities for Bessel functions. J. Comput. Appl. Math. 15 (1), pp. 75–81.

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External Links:

ISSN 0377-0427, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§10.14.

Encodings:

BibTeX

►

Bounds for modified Bessel functions. J. Comput. Appl. Math. 34 (3), pp. 263–267.

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External Links:

ISSN 0377-0427, Document, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.37.

Encodings:

BibTeX

… ►

Monotonicity of the differences of zeros of Bessel functions as a function of order. Proc. Amer. Math. Soc. 15 (1), pp. 91–96.

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External Links:

ISSN 0002-9939, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(ii).

Encodings:

BibTeX

…

45: 15.17 Mathematical Applications

… ►For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). …Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. …

46: 12.2 Differential Equations

… ►Standard solutions are $U(a,\pm z)$, $V(a,\pm z)$, $\overline{U}(a,\pm x)$ (not complex conjugate), $U(-a,\pm \mathrm{i}z)$ for (12.2.2); $W(a,\pm x)$ for (12.2.3); $D_{\nu }\left(\pm z\right)$ for (12.2.4), where ►

12.2.5 $$D_{\nu }\left(z\right)=U(-\frac{1}{2}-\nu ,z).$$

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Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $U(a,z)$: parabolic cylinder function, $z$: complex variable and $\nu$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.2.E5

Encodings:

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See also:

Annotations for §12.2(i), §12.2 and Ch.12

►All solutions are entire functions of $z$ and entire functions of $a$ or $\nu$. … ►

§12.2(vi) Solution $\overline{U}(a,x)$; Modulus and Phase Functions

►When $z$ $(=x)$ is real the solution $\overline{U}(a,x)$ is defined by …

47: 1.4 Calculus of One Variable

… ►A function $f(x)$ is continuous on the right (or from above) at $x=c$ if … ►A more general concept of integrability of a function on a bounded or unbounded interval is Lebesgue integrability, which allows discussion of functions which may not be well defined everywhere (especially on sets of measure zero) for $x\in ℝ$. … ►A function $f(x)$ is square-integrable if … ►This definition also applies when $f(x)$ is a complex function of the real variable $x$. … ►A function $f(x)$ is convex on $(a,b)$ if …

48: 13.6 Relations to Other Functions

… ►

13.6.3 $$M(0,b,z)=U(0,b,z)=1,$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.6.E3

Encodings:

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See also:

Annotations for §13.6(i), §13.6 and Ch.13

… ►When $a-b$ is an integer or $a$ is a positive integer the Kummer functions can be expressed as incomplete gamma functions (or generalized exponential integrals). … ►When $b=2a$ the Kummer functions can be expressed as modified Bessel functions. … ►

13.6.11_1 $$M(\nu +\frac{1}{2},2\nu +1+n,2z)=\mathrm{\Gamma }\left(\nu \right){\mathrm{e}}^z{\left(z/2\right)}^{-\nu }\sum _{k=0}^n\frac{{\left(-n\right)}_k{\left(2\nu \right)}_k(\nu +k)}{{\left(2\nu +1+n\right)}_{k}k!}{I}_{\nu +k}\left(z\right),$$

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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, $n$: nonnegative integer and $z$: complex variable

Source:

Prudnikov et al. (1990, page 579, (7.11.1.6))

Referenced by:

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

Permalink:

http://dlmf.nist.gov/13.6.E11_1

Encodings:

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Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.6(iii), §13.6 and Ch.13

… ►

13.6.20 $$U(-n,z-n+1,a)={\left(-z\right)}_{n}M(-n,z-n+1,a)=a^n{C}_n(z;a).$$

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Symbols:

$C_n(x;a)$: Charlier polynomial, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.6(v)

Permalink:

http://dlmf.nist.gov/13.6.E20

Encodings:

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See also:

Annotations for §13.6(v), §13.6(v), §13.6 and Ch.13

…

49: 36.13 Kelvin’s Ship-Wave Pattern

… ►

36.13.1 $$z(\varphi ,\rho )={\int }_{-\pi /2}^{\pi /2}\cos \left(\rho \frac{\cos \left(\theta +\varphi \right)}{{\cos }^2\theta }\right)d\theta ,$$

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Defines:

$z(\varphi ,\rho )$: wave height approximant (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ and $\int$: integral

Permalink:

http://dlmf.nist.gov/36.13.E1

Encodings:

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See also:

Annotations for §36.13 and Ch.36

… ►The wake is a caustic of the “rays” defined by the dispersion relation (“Hamiltonian”) giving the frequency $\omega$ as a function of wavevector $𝐤$: … ►

36.13.8 $$z(\rho ,\varphi )=2\pi \left({\rho }^{-1/3}u(\varphi )\cos \left(\rho \tilde{f}(\varphi )\right)\mathrm{Ai}\left(-{\rho }^{2/3}\mathrm{\Delta }(\varphi )\right)(1+O\left(1/\rho \right))+{\rho }^{-2/3}v(\varphi )\sin \left(\rho \tilde{f}(\varphi )\right){\mathrm{Ai}}^{\prime }\left(-{\rho }^{2/3}\mathrm{\Delta }(\varphi )\right)(1+O\left(1/\rho \right))\right),$$ $\rho \to \mathrm{\infty }$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z(\varphi ,\rho )$: wave height approximant, $u(\varphi )$: function and $v(\varphi )$: function

Referenced by:

Figure 36.13.1, Figure 36.13.1

Permalink:

http://dlmf.nist.gov/36.13.E8

Encodings:

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See also:

Annotations for §36.13 and Ch.36

… ►

►►

Figure 36.13.1: Kelvin’s ship wave pattern, computed from the uniform asymptotic approximation (36.13.8), as a function of $x=\rho \cos \varphi$, $y=\rho \sin \varphi$. Magnify

…

50: 15.2 Definitions and Analytical Properties

… ►The hypergeometric function $F(a,b;c;z)$ is defined by the Gauss series … ►The principal branch of $𝐅(a,b;c;z)$ is an entire function of $a$, $b$, and $c$. …As a multivalued function of $z$, $𝐅(a,b;c;z)$ is analytic everywhere except for possible branch points at $z=0$, $1$, and $\mathrm{\infty }$. The same properties hold for $F(a,b;c;z)$, except that as a function of $c$, $F(a,b;c;z)$ in general has poles at $c=0,-1,-2,\mathrm{\dots }$. … ►For example, when $a=-m$, $m=0,1,2,\mathrm{\dots }$, and $c\ne 0,-1,-2,\mathrm{\dots }$, $F(a,b;c;z)$ is a polynomial: …