Kummer solutions

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1—10 of 20 matching pages

1: 15.10 Hypergeometric Differential Equation

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§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

►The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions. … ►The

\genfrac{(}{)}{0.0pt}{}{6}{3}=20

connection formulas for the principal branches of Kummer’s solutions are: … ►

15.10.21

w_{1}(z)=\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a% \right)\Gamma\left(c-b\right)}w_{3}(z)+\frac{\Gamma\left(c\right)\Gamma\left(a% +b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}w_{4}(z),

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $w_{1}(z)$ : Kummer’s solution $w_{1}$ , $w_{3}(z)$ : Kummer’s solution $w_{3}$ and $w_{4}(z)$ : Kummer’s solution $w_{4}$
Referenced by:: §15.8(i)
Permalink:: http://dlmf.nist.gov/15.10.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.25

w_{1}(z)=\frac{\Gamma\left(c\right)\Gamma\left(b-a\right)}{\Gamma\left(b\right% )\Gamma\left(c-a\right)}w_{5}(z)+\frac{\Gamma\left(c\right)\Gamma\left(a-b% \right)}{\Gamma\left(a\right)\Gamma\left(c-b\right)}w_{6}(z),

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $w_{1}(z)$ : Kummer’s solution $w_{1}$ , $w_{5}(z)$ : Kummer’s solution $w_{5}$ and $w_{6}(z)$ : Kummer’s solution $w_{6}$
Referenced by:: §15.8(i)
Permalink:: http://dlmf.nist.gov/15.10.E25
Encodings:: pMML, png, TeX
See also:: Annotations for §15.10(ii), §15.10 and Ch.15

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2: 31.11 Expansions in Series of Hypergeometric Functions

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31.11.3_1

P_{j}^{5}=\frac{{\left(\lambda\right)_{j}}{\left(1-\gamma+\lambda\right)_{j}}}% {{\left(1+\lambda-\mu\right)_{2j}}}z^{-\lambda-j}\*{{}_{2}F_{1}}\left({\lambda% +j,1-\gamma+\lambda+j\atop 1+\lambda-\mu+2j};\frac{1}{z}\right),

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Defines:: $P_{j}^{5}$ : first Kummer solution at $\infty$ of generalized hypergeometric differential equation (locally)
Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $\lambda$ and $\mu$
Referenced by:: §31.11(ii), Erratum (V1.1.7) for Additions
Permalink:: http://dlmf.nist.gov/31.11.E3_1
Encodings:: pMML, png, TeX
Addition (effective with 1.1.7):: This equation was added.
Suggested 2022-09-14 by Hans Volkmer
See also:: Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.3_2

P_{j}^{6}=\frac{{\left(\lambda-\mu\right)_{2j}}}{{\left(1-\mu\right)_{j}}{% \left(\gamma-\mu\right)_{j}}}z^{-\mu+j}\*{{}_{2}F_{1}}\left({\mu-j,1-\gamma+% \mu-j\atop 1-\lambda+\mu-2j};\frac{1}{z}\right).

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Defines:: $P_{j}^{6}$ : second Kummer solution at $\infty$ of generalized hypergeometric differential equation (locally)
Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $\lambda$ and $\mu$
Referenced by:: §31.11(ii), Erratum (V1.1.7) for Additions
Permalink:: http://dlmf.nist.gov/31.11.E3_2
Encodings:: pMML, png, TeX
Addition (effective with 1.1.7):: This equation was added.
Suggested 2022-09-14 by Hans Volkmer
See also:: Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.12

P_{j}^{5}=\frac{{\left(\alpha\right)_{j}}{\left(1-\gamma+\alpha\right)_{j}}}{{% \left(1+\alpha-\beta+\epsilon\right)_{2j}}}z^{-\alpha-j}\*{{}_{2}F_{1}}\left({% \alpha+j,1-\gamma+\alpha+j\atop 1+\alpha-\beta+\epsilon+2j};\frac{1}{z}\right),

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $z$ : complex variable, $\gamma$ : real or complex parameter, $\epsilon$ : real or complex parameter, $j$ : nonnegative integer, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter and $P_{j}^{5}$ : first Kummer solution at $\infty$ of generalized hypergeometric differential equation
Referenced by:: §31.11(iii), §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Subsection 31.11(iii)
Permalink:: http://dlmf.nist.gov/31.11.E12
Encodings:: pMML, png, TeX
Notational Change (effective with 1.1.7):: We have rewritten the gamma functions in the prefactor more concisely using Pochhammer symbols.
See also:: Annotations for §31.11(iii), §31.11 and Ch.31

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3: 13.2 Definitions and Basic Properties

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Standard Solutions

►The first two standard solutions are: … ►Another standard solution of (13.2.1) is

U\left(a,b,z\right)

, which is determined uniquely by the property … ►

§13.2(v) Numerically Satisfactory Solutions

… ► …

4: 12.7 Relations to Other Functions

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12.7.12

u_{1}(a,z)=e^{-\tfrac{1}{4}z^{2}}M\left(\tfrac{1}{2}a+\tfrac{1}{4},\tfrac{1}{2% },\tfrac{1}{2}z^{2}\right)=e^{\tfrac{1}{4}z^{2}}M\left(-\tfrac{1}{2}a+\tfrac{1% }{4},\tfrac{1}{2},-\tfrac{1}{2}z^{2}\right),

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Defines:: $u_{1}(a,z)$ : solution (locally)
Symbols:: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.2.2 (modification of)
Referenced by:: §12.20
Permalink:: http://dlmf.nist.gov/12.7.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §12.7(iv), §12.7 and Ch.12

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12.7.13

u_{2}(a,z)=ze^{-\tfrac{1}{4}z^{2}}M\left(\tfrac{1}{2}a+\tfrac{3}{4},\tfrac{3}{% 2},\tfrac{1}{2}z^{2}\right)=ze^{\tfrac{1}{4}z^{2}}M\left(-\tfrac{1}{2}a+\tfrac% {3}{4},\tfrac{3}{2},-\tfrac{1}{2}z^{2}\right).

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Defines:: $u_{2}(a,z)$ : solution (locally)
Symbols:: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.2.4 (modification of)
Referenced by:: §12.20
Permalink:: http://dlmf.nist.gov/12.7.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §12.7(iv), §12.7 and Ch.12

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5: 13.14 Definitions and Basic Properties

… ►Standard solutions are: …

6: 13.29 Methods of Computation

… ►The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. In Allasia and Besenghi (1991) and Allasia and Besenghi (1987a) the high accuracy of the trapezoidal rule for the computation of Kummer functions is described. … ►with recessive solution … ►with recessive solution … ►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of

M\left(n,b,x\right)

, when

b

and

x

are real and

n

is a positive integer. …

7: Errata

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Equation (13.2.7)

13.2.7

U\left(-m,b,z\right)=(-1)^{m}{\left(b\right)_{m}}M\left(-m,b,z\right)=(-1)^{m}% \sum_{s=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{s}{\left(b+s\right)_{m-s}}(-z)^{s}

The equality $U\left(-m,b,z\right)=(-1)^{m}{\left(b\right)_{m}}M\left(-m,b,z\right)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation $a=-n$ has been changed to $a=-m$ .

Reported 2015-02-10 by Adri Olde Daalhuis.

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Equation (13.2.8)

13.2.8

U\left(a,a+n+1,z\right)=\frac{(-1)^{n}{\left(1-a-n\right)_{n}}}{z^{a+n}}\*M% \left(-n,1-a-n,z\right)=z^{-a}\sum_{s=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{s}{% \left(a\right)_{s}}z^{-s}

The equality $U\left(a,a+n+1,z\right)=\frac{(-1)^{n}{\left(1-a-n\right)_{n}}}{z^{a+n}}\*M% \left(-n,1-a-n,z\right)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation.

Reported 2015-02-10 by Adri Olde Daalhuis.

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Equation (13.2.10)

13.2.10

U\left(-m,n+1,z\right)=(-1)^{m}{\left(n+1\right)_{m}}M\left(-m,n+1,z\right)=(-% 1)^{m}\sum_{s=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{s}{\left(n+s+1\right)_{m-s}}(-z% )^{s}

The equality $U\left(-m,n+1,z\right)=(-1)^{m}{\left(n+1\right)_{m}}\*M\left(-m,n+1,z\right)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation $a=-m,m=0,1,2,\ldots$ has been introduced.

Reported 2015-02-10 by Adri Olde Daalhuis.

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8: 31.3 Basic Solutions

… ►The full set of 192 local solutions of (31.2.1), equivalent in 8 sets of 24, resembles Kummer’s set of 24 local solutions of the hypergeometric equation, which are equivalent in 4 sets of 6 solutions (§15.10(ii)); see Maier (2007).

9: 13.28 Physical Applications

§13.28 Physical Applications

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§13.28(i) Exact Solutions of the Wave Equation

… ►and

V^{(j)}_{\kappa,\mu}(z)

j=1,2

, denotes any pair of solutions of Whittaker’s equation (13.14.1). …

10: 12.14 The Function $W\left(a,x\right)$

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12.14.15

w_{1}(a,x)=e^{-\frac{1}{4}ix^{2}}M\left(\tfrac{1}{4}-\tfrac{1}{2}ia,\tfrac{1}{% 2},\tfrac{1}{2}ix^{2}\right)=e^{\frac{1}{4}ix^{2}}M\left(\tfrac{1}{4}+\tfrac{1% }{2}ia,\tfrac{1}{2},-\tfrac{1}{2}ix^{2}\right),

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Symbols:: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $x$ : real variable, $a$ : real or complex parameter and $w_{1}(a,x)$ : even solution
A&S Ref:: 19.25.1 (modification of)
Referenced by:: §12.14(v)
Permalink:: http://dlmf.nist.gov/12.14.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §12.14(vii), §12.14(vii), §12.14 and Ch.12

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12.14.16

w_{2}(a,x)=xe^{-\frac{1}{4}ix^{2}}M\left(\tfrac{3}{4}-\tfrac{1}{2}ia,\tfrac{3}% {2},\tfrac{1}{2}ix^{2}\right)=xe^{\frac{1}{4}ix^{2}}M\left(\tfrac{3}{4}+\tfrac% {1}{2}ia,\tfrac{3}{2},-\tfrac{1}{2}ix^{2}\right).

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Symbols:: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $x$ : real variable, $a$ : real or complex parameter and $w_{2}(a,x)$ : odd solution
A&S Ref:: 19.25.1 (modification of)
Referenced by:: §12.14(v)
Permalink:: http://dlmf.nist.gov/12.14.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §12.14(vii), §12.14(vii), §12.14 and Ch.12

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Kummer solutions

1—10 of 20 matching pages

1: 15.10 Hypergeometric Differential Equation

§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

2: 31.11 Expansions in Series of Hypergeometric Functions

3: 13.2 Definitions and Basic Properties

Standard Solutions

§13.2(v) Numerically Satisfactory Solutions

4: 12.7 Relations to Other Functions

5: 13.14 Definitions and Basic Properties

6: 13.29 Methods of Computation

7: Errata

8: 31.3 Basic Solutions

9: 13.28 Physical Applications

§13.28 Physical Applications

§13.28(i) Exact Solutions of the Wave Equation

10: 12.14 The Function W ⁡ ( a , x )

10: 12.14 The Function $W\left(a,x\right)$