relation to Kummer equation

SearchAdvancedHelp

(0.004 seconds)

1—10 of 38 matching pages

1: 13.14 Definitions and Basic Properties

… ►

Whittaker’s Equation

… ►Standard solutions are: …

2: 13.2 Definitions and Basic Properties

… ►

13.2.1 $$z\frac{d^{2}w}{{dz}^2}+(b-z)\frac{dw}{dz}-aw=0.$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

A&S Ref:

13.1.1

Referenced by:

§13.14(i), §13.14(v), §13.2(i), §13.2(v), §13.29(ii), §13.3(i), §13.3(i)

Permalink:

http://dlmf.nist.gov/13.2.E1

Encodings:

pMML, png, TeX

See also:

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

…

3: 6.11 Relations to Other Functions

… ►

6.11.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}U(1,1,z),$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $z$: complex variable

Referenced by:

§6.11, 5th item, §6.6

Permalink:

http://dlmf.nist.gov/6.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

►

6.11.3 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)=U(1,1,-\mathrm{i}z).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.11, 5th item

Permalink:

http://dlmf.nist.gov/6.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

4: 13.6 Relations to Other Functions

… ►

§13.6(i) Elementary Functions

… ►

§13.6(ii) Incomplete Gamma Functions

… ►

§13.6(iv) Parabolic Cylinder Functions

… ►

§13.6(v) Orthogonal Polynomials

… ►

§13.6(vi) Generalized Hypergeometric Functions

…

5: 13.3 Recurrence Relations and Derivatives

§13.3 Recurrence Relations and Derivatives

►

§13.3(i) Recurrence Relations

… ►Kummer’s differential equation (13.2.1) is equivalent to … ►

§13.3(ii) Differentiation Formulas

… ►

13.3.22 $$\frac{d}{dz}U(a,b,z)=-aU(a+1,b+1,z),$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

A&S Ref:

13.4.21

Permalink:

http://dlmf.nist.gov/13.3.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(ii), §13.3 and Ch.13

…

6: 10.39 Relations to Other Functions

§10.39 Relations to Other Functions

►

Elementary Functions

… ►

Parabolic Cylinder Functions

… ►

Confluent Hypergeometric Functions

… ►

Generalized Hypergeometric Functions and Hypergeometric Function

…

7: 7.11 Relations to Other Functions

§7.11 Relations to Other Functions

►

Incomplete Gamma Functions and Generalized Exponential Integral

… ►

Confluent Hypergeometric Functions

… ►

7.11.4 $$\mathrm{erf}z=\frac{2z}{\sqrt{\pi }}M(\frac{1}{2},\frac{3}{2},-z^2)=\frac{2z}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}M(1,\frac{3}{2},z^2),$$

ⓘ

Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

7.1.21

Permalink:

http://dlmf.nist.gov/7.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

… ►

Generalized Hypergeometric Functions

…

8: 8.19 Generalized Exponential Integral

… ►

§8.19(i) Definition and Integral Representations

… ►In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase. … ►

§8.19(v) Recurrence Relation and Derivatives

… ►

§8.19(vi) Relation to Confluent Hypergeometric Function

… ►For $U(a,b,z)$ see §13.2(i). …

9: 10.16 Relations to Other Functions

§10.16 Relations to Other Functions

►

Elementary Functions

… ►

Parabolic Cylinder Functions

… ►

Confluent Hypergeometric Functions

… ►

Generalized Hypergeometric Functions

…

10: 18.11 Relations to Other Functions

§18.11 Relations to Other Functions

… ►See §§18.5(i) and 18.5(iii) for relations to trigonometric functions, the hypergeometric function, and generalized hypergeometric functions. ►

Ultraspherical

… ►

Laguerre

… ►

Hermite

…