13 Confluent Hypergeometric Functions Kummer Functions 13.1 Special Notation 13.3 Recurrence Relations and Derivatives

§13.2 Definitions and Basic Properties

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Keywords:: Kummer functions, definitions
Referenced by:: §33.22(ii), §9.10(v), §9.6(iii)
Permalink:: http://dlmf.nist.gov/13.2
See also:: Annotations for Ch.13

§13.2(i) Differential Equation
§13.2(ii) Analytic Continuation
§13.2(iii) Limiting Forms as $z\to 0$
§13.2(iv) Limiting Forms as $z\to\infty$
§13.2(v) Numerically Satisfactory Solutions
§13.2(vi) Wronskians
§13.2(vii) Connection Formulas

§13.2(i) Differential Equation

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Notes:: See Olver (1997b, Chapter 7, §§3, 9, 10) and Temme (1996b, §§7.1–7.2). (13.2.7) and (13.2.8) are terminating forms of the asymptotic expansion (13.7.3) (that the $\sim$ sign can be replaced by $=$ in these circumstances follows from (13.7.4) and (13.7.5).)
Referenced by:: §10.16, §10.22(iv), §10.39, §12.14(vii), §12.20, §12.7(iv), §13.6(vi), §18.11(i), §18.30(iii), §31.12, §33.14(ii), §33.2(ii), §33.2(iii), §6.11, 5th item, §7.11, §7.18(iv), §8.19(vi), §8.5, Notation for the Special Functions
Permalink:: http://dlmf.nist.gov/13.2.i
See also:: Annotations for §13.2 and Ch.13

Kummer’s Equation

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Keywords:: Kummer functions, Kummer’s equation, integer parameters, relation to hypergeometric differential equation
See also:: Annotations for §13.2(i), §13.2 and Ch.13

13.2.1		$z\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+(b-z)\frac{\mathrm{d}w}{\mathrm{d% }z}-aw=0.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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: TeX, pMML, png See also: Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

This equation has a regular singularity at the origin with indices $0$ and $1-b$ , and an irregular singularity at infinity of rank one. It can be regarded as the limiting form of the hypergeometric differential equation (§15.10(i)) that is obtained on replacing $z$ by $\ifrac{z}{b}$ , letting $b\to\infty$ , and subsequently replacing the symbol $c$ by $b$ . In effect, the regular singularities of the hypergeometric differential equation at $b$ and $\infty$ coalesce into an irregular singularity at $\infty$ .

Standard Solutions

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Keywords:: Kummer functions, Kummer’s equation, Maclaurin, Maclaurin series, analytical properties, integer parameters, interrelations, polynomial cases, principal branches (or values), series expansions, standard solutions
See also:: Annotations for §13.2(i), §13.2 and Ch.13

The first two standard solutions are:

13.2.2		$M\left(a,b,z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}}{{\left(b% \right)_{s}}s!}z^{s}=1+\frac{a}{b}z+\frac{a(a+1)}{b(b+1)2!}z^{2}+\cdots,$
ⓘ Defines: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function Symbols: ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer and $z$ : complex variable A&S Ref: 13.1.2 Referenced by: §13.14(i), §13.2(i), §13.29(i), §13.29(ii), §13.9(i) Permalink: http://dlmf.nist.gov/13.2.E2 Encodings: TeX, pMML, png See also: Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

and

13.2.3		${\mathbf{M}}\left(a,b,z\right)=\sum_{s=0}^{\infty}\frac{{\left(a\right)_{s}}}{% \Gamma\left(b+s\right)s!}z^{s},$
ⓘ Defines: ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer and $z$ : complex variable Referenced by: §13.2(i) Permalink: http://dlmf.nist.gov/13.2.E3 Encodings: TeX, pMML, png See also: Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

except that $M\left(a,b,z\right)$ does not exist when $b$ is a nonpositive integer. In other cases

13.2.4		$M\left(a,b,z\right)=\Gamma\left(b\right){\mathbf{M}}\left(a,b,z\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function and $z$ : complex variable Referenced by: §13.2(ii) Permalink: http://dlmf.nist.gov/13.2.E4 Encodings: TeX, pMML, png See also: Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

The series (13.2.2) and (13.2.3) converge for all $z\in\mathbb{C}$ . $M\left(a,b,z\right)$ is entire in $z$ and $a$ , and is a meromorphic function of $b$ . ${\mathbf{M}}\left(a,b,z\right)$ is entire in $z$ , $a$ , and $b$ .

Although $M\left(a,b,z\right)$ does not exist when $b=-n$ , $n=0,1,2,\dots$ , many formulas containing $M\left(a,b,z\right)$ continue to apply in their limiting form. In particular,

13.2.5		$\lim_{b\to-n}\frac{M\left(a,b,z\right)}{\Gamma\left(b\right)}={\mathbf{M}}% \left(a,-n,z\right)=\frac{{\left(a\right)_{n+1}}}{(n+1)!}z^{n+1}M\left(a+n+1,n% +2,z\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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 Encodings: TeX, pMML, png See also: Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

When $a=-n$ , $n=0,1,2,\dots$ , ${\mathbf{M}}\left(a,b,z\right)$ is a polynomial in $z$ of degree not exceeding $n$ ; this is also true of $M\left(a,b,z\right)$ provided that $b$ is not a nonpositive integer.

Another standard solution of (13.2.1) is $U\left(a,b,z\right)$ , which is determined uniquely by the property

13.2.6		$U\left(a,b,z\right)\sim z^{-a},$
$z\to\infty$ , $\|\operatorname{ph}z\|\leq\frac{3}{2}\pi-\delta$ ,
ⓘ Defines: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function Symbols: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $z$ : complex variable and $\delta$ : small positive constant Referenced by: §13.2(i), §13.2(iv) Permalink: http://dlmf.nist.gov/13.2.E6 Encodings: TeX, pMML, png See also: Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

where $\delta$ is an arbitrary small positive constant. In general, $U\left(a,b,z\right)$ has a branch point at $z=0$ . The principal branch corresponds to the principal value of $z^{-a}$ in (13.2.6), and has a cut in the $z$ -plane along the interval $(-\infty,0]$ ; compare §4.2(i).

When $a=-m$ , $m=0,1,2,\dots$ , $U\left(a,b,z\right)$ is a polynomial in $z$ of degree $m$ :

13.2.7		$U\left(-m,b,z\right)=(-1)^{m}{\left(b\right)_{m}}M\left(-m,b,z\right)=(-1)^{m}% \sum_{s=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{s}{\left(b+s\right)_{m-s}}(-z)^{s}.$
ⓘ 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, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $m$ : integer, $n$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable Referenced by: §13.2(i), Erratum (V1.0.10) for Equation (13.2.7) Permalink: http://dlmf.nist.gov/13.2.E7 Encodings: TeX, pMML, png Addition (effective with 1.0.10): 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 $a=-m$ . See also: Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

Similarly, when $a-b+1=-n$ , $n=0,1,2,\ldots$ ,

13.2.8		$U\left(a,a+n+1,z\right)=\frac{(-1)^{n}{\left(1-a-n\right)_{n}}}{z^{a+n}}M\left% (-n,1-a-n,z\right)=z^{-a}\sum_{s=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{s}{\left(a% \right)_{s}}z^{-s}.$
ⓘ 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, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $n$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable Referenced by: §13.2(i), Erratum (V1.0.10) for Equation (13.2.8) Permalink: http://dlmf.nist.gov/13.2.E8 Encodings: TeX, pMML, png Addition (effective with 1.0.10): 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. Suggested 2014-02-10 by Adri Olde Daalhuis See also: Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

When $b=n+1$ , $n=0,1,2,\dots$ , and $a\neq 0,-1,-2,\dots$ ,

13.2.9		$U\left(a,n+1,z\right)=\frac{(-1)^{n+1}}{n!\Gamma\left(a-n\right)}\sum_{k=0}^{% \infty}\frac{{\left(a\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\ln z+% \psi\left(a+k\right)-\psi\left(1+k\right)-\psi\left(n+k+1\right)\right)+\frac{% 1}{\Gamma\left(a\right)}\sum_{k=1}^{n}\frac{(k-1)!{\left(1-a+k\right)_{n-k}}}{% (n-k)!}z^{-k}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §13.2(i), §33.6, §6.6, Erratum (V1.0.12) for Equations (13.2.9), (13.2.10) Permalink: http://dlmf.nist.gov/13.2.E9 Encodings: TeX, pMML, png Clarification (effective with 1.0.12): The condition $a\neq 0,-1,-2,\dots$ that appeared below this equation now appears ahead of the equation. Suggested 2016-07-05 by Adri Olde Daalhuis See also: Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

When $b=n+1$ , $n=0,1,2,\dots$ , and $a=-m$ , $m=0,1,2,\dots$ ,

13.2.10		$U\left(-m,n+1,z\right)=(-1)^{m}{\left(n+1\right)_{m}}M\left(-m,n+1,z\right)=(-% 1)^{m}\sum_{s=0}^{m}\genfrac{(}{)}{0.0pt}{}{m}{s}{\left(n+s+1\right)_{m-s}}(-z% )^{s}.$
ⓘ 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, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $m$ : integer, $n$ : nonnegative integer, $s$ : nonnegative integer and $z$ : complex variable Referenced by: §13.2(i), Erratum (V1.0.10) for Equation (13.2.10), Erratum (V1.0.12) for Equations (13.2.9), (13.2.10) Permalink: http://dlmf.nist.gov/13.2.E10 Encodings: TeX, pMML, png Clarification (effective with 1.0.12): . The condition $a=-m,m=0,1,2,\dots$ that appeared below this equation now appears ahead of the equation. Suggested 2016-07-05 by Adri Olde Daalhuis Addition (effective with 1.0.10): 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. Suggested 2015-02-10 by Adri Olde Daalhuis See also: Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

When $b=-n$ , $n=0,1,2,\dots$ , the following equation can be combined with (13.2.9) and (13.2.10):

13.2.11		$U\left(a,-n,z\right)=z^{n+1}U\left(a+n+1,n+2,z\right).$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.2.E11 Encodings: TeX, pMML, png See also: Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

§13.2(ii) Analytic Continuation

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Keywords:: Kummer functions, analytic continuation
Notes:: To verify (13.2.12) replace the $U$ functions by ${\mathbf{M}}$ functions by means of (13.2.42) and (13.2.4), then recall that each ${\mathbf{M}}$ function is an entire function of $z$ .
Permalink:: http://dlmf.nist.gov/13.2.ii
See also:: Annotations for §13.2 and Ch.13

When $m\in\mathbb{Z}$ ,

13.2.12		$U\left(a,b,ze^{2\pi\mathrm{i}m}\right)=\frac{2\pi\mathrm{i}e^{-\pi\mathrm{i}bm% }\sin\left(\pi bm\right)}{\Gamma\left(1+a-b\right)\sin\left(\pi b\right)}{% \mathbf{M}}\left(a,b,z\right)+e^{-2\pi\mathrm{i}bm}U\left(a,b,z\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $m$ : integer and $z$ : complex variable A&S Ref: 13.1.10 (in different form) Referenced by: §13.2(ii), §7.18(iv) Permalink: http://dlmf.nist.gov/13.2.E12 Encodings: TeX, pMML, png See also: Annotations for §13.2(ii), §13.2 and Ch.13

Except when $z=0$ each branch of $U\left(a,b,z\right)$ is entire in $a$ and $b$ . Unless specified otherwise, however, $U\left(a,b,z\right)$ is assumed to have its principal value.

§13.2(iii) Limiting Forms as $z\to 0$

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Keywords:: Kummer functions, as $z\rightarrow 0$ , limiting forms
Notes:: See Temme (1996b, Ex. 7.6: an error in the equation that corresponds to (13.2.19) has been corrected).
Referenced by:: §13.14(iii)
Permalink:: http://dlmf.nist.gov/13.2.iii
See also:: Annotations for §13.2 and Ch.13

13.2.13		$M\left(a,b,z\right)=1+O\left(z\right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.2.E13 Encodings: TeX, pMML, png See also: Annotations for §13.2(iii), §13.2 and Ch.13

Next, in cases when $a=-n$ or $-n+b-1$ , where $n$ is a nonnegative integer,

13.2.14		$U\left(-n,b,z\right)=(-1)^{n}{\left(b\right)_{n}}+O\left(z\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.2.E14 Encodings: TeX, pMML, png See also: Annotations for §13.2(iii), §13.2 and Ch.13

13.2.15		$U\left(-n+b-1,b,z\right)=(-1)^{n}{\left(2-b\right)_{n}}z^{1-b}+O\left(z^{2-b}% \right).$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.2.E15 Encodings: TeX, pMML, png See also: Annotations for §13.2(iii), §13.2 and Ch.13

In all other cases

13.2.16	$\displaystyle U\left(a,b,z\right)$	$\displaystyle=\frac{\Gamma\left(b-1\right)}{\Gamma\left(a\right)}z^{1-b}+O% \left(z^{2-\Re b}\right),$
$\Re b\geq 2$ , $b\not=2$ ,
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\Re$ : real part and $z$ : complex variable A&S Ref: 13.5.6 (with order estimate corrected) Permalink: http://dlmf.nist.gov/13.2.E16 Encodings: TeX, pMML, png See also: Annotations for §13.2(iii), §13.2 and Ch.13
13.2.17	$\displaystyle U\left(a,2,z\right)$	$\displaystyle=\frac{1}{\Gamma\left(a\right)}z^{-1}+O\left(\ln z\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 13.5.7 Permalink: http://dlmf.nist.gov/13.2.E17 Encodings: TeX, pMML, png See also: Annotations for §13.2(iii), §13.2 and Ch.13
13.2.18	$\displaystyle U\left(a,b,z\right)$	$\displaystyle=\frac{\Gamma\left(b-1\right)}{\Gamma\left(a\right)}z^{1-b}+\frac% {\Gamma\left(1-b\right)}{\Gamma\left(a-b+1\right)}+O\left(z^{2-\Re b}\right),$
$1\leq\Re b<2$ , $b\not=1$ ,
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\Re$ : real part and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.2.E18 Encodings: TeX, pMML, png See also: Annotations for §13.2(iii), §13.2 and Ch.13
13.2.19	$\displaystyle U\left(a,1,z\right)$	$\displaystyle=-\frac{1}{\Gamma\left(a\right)}\left(\ln z+\psi\left(a\right)+2% \gamma\right)+O\left(z\ln z\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\gamma$ : Euler’s constant, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 13.5.9 (as corrected in later editions) Referenced by: §13.2(iii) Permalink: http://dlmf.nist.gov/13.2.E19 Encodings: TeX, pMML, png See also: Annotations for §13.2(iii), §13.2 and Ch.13
13.2.20	$\displaystyle U\left(a,b,z\right)$	$\displaystyle=\frac{\Gamma\left(1-b\right)}{\Gamma\left(a-b+1\right)}+O\left(z% ^{1-\Re b}\right),$
$0<\Re b<1$ ,
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\Re$ : real part and $z$ : complex variable A&S Ref: 13.5.10 Permalink: http://dlmf.nist.gov/13.2.E20 Encodings: TeX, pMML, png See also: Annotations for §13.2(iii), §13.2 and Ch.13
13.2.21	$\displaystyle U\left(a,0,z\right)$	$\displaystyle=\frac{1}{\Gamma\left(a+1\right)}+O\left(z\ln z\right),$
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 13.5.11 Permalink: http://dlmf.nist.gov/13.2.E21 Encodings: TeX, pMML, png See also: Annotations for §13.2(iii), §13.2 and Ch.13
13.2.22	$\displaystyle U\left(a,b,z\right)$	$\displaystyle=\frac{\Gamma\left(1-b\right)}{\Gamma\left(a-b+1\right)}+O\left(z% \right),$
$\Re b\leq 0$ , $b\not=0$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\Re$ : real part and $z$ : complex variable A&S Ref: 13.5.12 Permalink: http://dlmf.nist.gov/13.2.E22 Encodings: TeX, pMML, png See also: Annotations for §13.2(iii), §13.2 and Ch.13

§13.2(iv) Limiting Forms as $z\to\infty$

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Keywords:: Kummer functions, as $z\rightarrow\infty$ , limiting forms
Notes:: See (13.7.2).
Permalink:: http://dlmf.nist.gov/13.2.iv
See also:: Annotations for §13.2 and Ch.13

Except when $a=0,-1,\dots$ (polynomial cases),

13.2.23		${\mathbf{M}}\left(a,b,z\right)\sim\ifrac{e^{z}z^{a-b}}{\Gamma\left(a\right)},$
$\left\|\operatorname{ph}z\right\|\leq\frac{1}{2}\pi-\delta$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $z$ : complex variable and $\delta$ : small positive constant Permalink: http://dlmf.nist.gov/13.2.E23 Encodings: TeX, pMML, png See also: Annotations for §13.2(iv), §13.2 and Ch.13

where $\delta$ is an arbitrary small positive constant.

For $U\left(a,b,z\right)$ see (13.2.6).

§13.2(v) Numerically Satisfactory Solutions

ⓘ

Keywords:: Kummer’s equation, fundamental solutions, numerically satisfactory solutions
Notes:: See Olver (1997b, Chapter 7, §§9, 10). (13.2.27)–(13.2.32) are obtained by considering limiting forms of the connection formulas in §13.2(vii); see Olver (1997b, Chapter 7, Ex. 10.6) and Slater (1960, §§1.5–1.5.1).
Referenced by:: §13.14(v)
Permalink:: http://dlmf.nist.gov/13.2.v
See also:: Annotations for §13.2 and Ch.13

Fundamental pairs of solutions of (13.2.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are

13.2.24		$U\left(a,b,z\right)$ ,
		$e^{z}U\left(b-a,b,e^{-\pi\mathrm{i}}z\right)$ ,
	$-\tfrac{1}{2}\pi\leq\operatorname{ph}{z}\leq\tfrac{3}{2}\pi$ ,
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.2.E24 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §13.2(v), §13.2 and Ch.13
13.2.25		$U\left(a,b,z\right)$ ,
		$e^{z}U\left(b-a,b,e^{\pi\mathrm{i}}z\right)$ ,
	$-\tfrac{3}{2}\pi\leq\operatorname{ph}{z}\leq\tfrac{1}{2}\pi$ .
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.2.E25 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §13.2(v), §13.2 and Ch.13

A fundamental pair of solutions that is numerically satisfactory near the origin is

13.2.26		$M\left(a,b,z\right),\quad z^{1-b}M\left(a-b+1,2-b,z\right),$
$b\not\in\mathbb{Z}$ .
ⓘ 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, $\mathbb{Z}$ : set of all integers, $\notin$ : not an element of and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.2.E26 Encodings: TeX, pMML, png See also: Annotations for §13.2(v), §13.2 and Ch.13

When $b=n+1=1,2,3,\dots$ , a fundamental pair that is numerically satisfactory near the origin is $M\left(a,n+1,z\right)$ and

13.2.27		$\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(1-a\right)_{k}}}z^{-k}-\sum_{k=0}^{% \infty}\frac{{\left(a\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\ln z+% \psi\left(a+k\right)-\psi\left(1+k\right)-\psi\left(n+k+1\right)\right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §13.2(v) Permalink: http://dlmf.nist.gov/13.2.E27 Encodings: TeX, pMML, png See also: Annotations for §13.2(v), §13.2 and Ch.13

if $a-n\neq 0,-1,-2,\dots$ , or $M\left(a,n+1,z\right)$ and

13.2.28		$\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(1-a\right)_{k}}}z^{-k}-\sum_{k=0}^{% -a}\frac{{\left(a\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\ln z+\psi% \left(1-a-k\right)-\psi\left(1+k\right)-\psi\left(n+k+1\right)\right)+(-1)^{1-% a}(-a)!\sum_{k=1-a}^{\infty}\frac{(k-1+a)!}{{\left(n+1\right)_{k}}k!}z^{k},$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.2.E28 Encodings: TeX, pMML, png See also: Annotations for §13.2(v), §13.2 and Ch.13

if $a=0,-1,-2,\dots$ , or $M\left(a,n+1,z\right)$ and

13.2.29		$\sum_{k=a}^{n}\frac{(k-1)!}{(n-k)!(k-a)!}z^{-k},$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.2.E29 Encodings: TeX, pMML, png See also: Annotations for §13.2(v), §13.2 and Ch.13

if $a=1,2,\dots,n$ .

When $b=-n=0,-1,-2,\dots$ , a fundamental pair that is numerically satisfactory near the origin is $z^{n+1}\*M\left(a+n+1,n+2,z\right)$ and

13.2.30		$\sum_{k=1}^{n+1}\frac{(n+1)!(k-1)!}{(n-k+1)!{\left(-a-n\right)_{k}}}z^{n-k+1}-% \sum_{k=0}^{\infty}\frac{{\left(a+n+1\right)_{k}}}{{\left(n+2\right)_{k}}k!}z^% {n+k+1}\left(\ln z+\psi\left(a+n+k+1\right)-\psi\left(1+k\right)-\psi\left(n+k% +2\right)\right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.2.E30 Encodings: TeX, pMML, png See also: Annotations for §13.2(v), §13.2 and Ch.13

if $a\neq 0,-1,-2,\dots$ , or $z^{n+1}M\left(a+n+1,n+2,z\right)$ and

13.2.31		$\sum_{k=1}^{n+1}\frac{(n+1)!(k-1)!}{(n-k+1)!{\left(-a-n\right)_{k}}}z^{n-k+1}-% \sum_{k=0}^{-a-n-1}\frac{{\left(a+n+1\right)_{k}}}{{\left(n+2\right)_{k}}k!}z^% {n+k+1}\left(\ln z+\psi\left(-a-n-k\right)-\psi\left(1+k\right)-\psi\left(n+k+% 2\right)\right)+(-1)^{n-a}{(-a-n-1)!}\sum_{k=-a-n}^{\infty}\frac{(k+a+n)!}{{% \left(n+2\right)_{k}}k!}z^{n+k+1},$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.2.E31 Encodings: TeX, pMML, png See also: Annotations for §13.2(v), §13.2 and Ch.13

if $a=-n-1,-n-2,-n-3,\dots$ , or $z^{n+1}M\left(a+n+1,n+2,z\right)$ and

13.2.32		$\sum_{k=a+n+1}^{n+1}\frac{(k-1)!}{(n-k+1)!(k-a-n-1)!}z^{n-k+1},$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $z$ : complex variable Referenced by: §13.2(v) Permalink: http://dlmf.nist.gov/13.2.E32 Encodings: TeX, pMML, png See also: Annotations for §13.2(v), §13.2 and Ch.13

if $a=0,-1,\dots,-n$ .

§13.2(vi) Wronskians

ⓘ

Keywords:: Kummer functions, Wronskians
Notes:: See Slater (1960, §2.1.2).
Permalink:: http://dlmf.nist.gov/13.2.vi
See also:: Annotations for §13.2 and Ch.13

13.2.33	$\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),z^{1-b}{\mathbf{% M}}\left(a-b+1,2-b,z\right)\right\}$	$\displaystyle=\sin\left(\pi b\right)z^{-b}e^{z}/\pi,$
ⓘ Symbols: ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 13.1.20 Permalink: http://dlmf.nist.gov/13.2.E33 Encodings: TeX, pMML, png See also: Annotations for §13.2(vi), §13.2 and Ch.13
13.2.34	$\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),U\left(a,b,z% \right)\right\}$	$\displaystyle=-\ifrac{z^{-b}e^{z}}{\Gamma\left(a\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\mathscr{W}$ : Wronskian, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable A&S Ref: 13.1.22 Permalink: http://dlmf.nist.gov/13.2.E34 Encodings: TeX, pMML, png See also: Annotations for §13.2(vi), §13.2 and Ch.13
13.2.35	$\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),e^{z}U\left(b-a,% b,e^{\pm\pi\mathrm{i}}z\right)\right\}$	$\displaystyle=\ifrac{e^{\mp b\pi\mathrm{i}}z^{-b}e^{z}}{\Gamma\left(b-a\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\mathscr{W}$ : 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: TeX, pMML, png See also: Annotations for §13.2(vi), §13.2 and Ch.13
13.2.36	$\displaystyle\mathscr{W}\left\{z^{1-b}{\mathbf{M}}\left(a-b+1,2-b,z\right),U% \left(a,b,z\right)\right\}$	$\displaystyle=-\ifrac{z^{-b}e^{z}}{\Gamma\left(a-b+1\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\mathscr{W}$ : Wronskian, $\mathrm{e}$ : base of natural logarithm and $z$ : complex variable A&S Ref: 13.1.24 Permalink: http://dlmf.nist.gov/13.2.E36 Encodings: TeX, pMML, png See also: Annotations for §13.2(vi), §13.2 and Ch.13
13.2.37	$\displaystyle\mathscr{W}\left\{z^{1-b}{\mathbf{M}}\left(a-b+1,2-b,z\right),e^{% z}U\left(b-a,b,e^{\pm\pi\mathrm{i}}z\right)\right\}$	$\displaystyle=-\ifrac{z^{-b}e^{z}}{\Gamma\left(1-a\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\mathscr{W}$ : 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: 13.1.25 Permalink: http://dlmf.nist.gov/13.2.E37 Encodings: TeX, pMML, png See also: Annotations for §13.2(vi), §13.2 and Ch.13
13.2.38	$\displaystyle\mathscr{W}\left\{U\left(a,b,z\right),e^{z}U\left(b-a,b,e^{\pm\pi% \mathrm{i}}z\right)\right\}$	$\displaystyle=e^{\pm(a-b)\pi\mathrm{i}}z^{-b}e^{z}.$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathscr{W}$ : 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: 13.1.26 Permalink: http://dlmf.nist.gov/13.2.E38 Encodings: TeX, pMML, png See also: Annotations for §13.2(vi), §13.2 and Ch.13

§13.2(vii) Connection Formulas

ⓘ

Keywords:: Kummer functions, connection formulas
Notes:: See Temme (1996b, §§7.1, 7.2).
Referenced by:: §13.2(v), §13.29(ii), §33.14(ii), §33.2(ii), §8.5
Permalink:: http://dlmf.nist.gov/13.2.vii
Addition (effective with 1.2.2):: Equations (13.2.43) and (13.2.44) were added.
See also:: Annotations for §13.2 and Ch.13

Kummer’s Transformations

ⓘ

Keywords:: Kummer functions, Kummer’s transformations, for confluent hypergeometric functions, interrelations
See also:: Annotations for §13.2(vii), §13.2 and Ch.13

13.2.39	$\displaystyle M\left(a,b,z\right)$	$\displaystyle=e^{z}M\left(b-a,b,-z\right),$
ⓘ 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 and $z$ : complex variable A&S Ref: 13.1.27 Referenced by: §13.12, §13.8(i), §13.9(ii), §18.17(vi) Permalink: http://dlmf.nist.gov/13.2.E39 Encodings: TeX, pMML, png See also: Annotations for §13.2(vii), §13.2(vii), §13.2 and Ch.13
13.2.40	$\displaystyle U\left(a,b,z\right)$	$\displaystyle=z^{1-b}U\left(a-b+1,2-b,z\right).$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and $z$ : complex variable A&S Ref: 13.1.29 Referenced by: §13.8(iii) Permalink: http://dlmf.nist.gov/13.2.E40 Encodings: TeX, pMML, png See also: Annotations for §13.2(vii), §13.2(vii), §13.2 and Ch.13

13.2.41		$\frac{1}{\Gamma\left(b\right)}M\left(a,b,z\right)=\frac{e^{\mp a\pi\mathrm{i}}% }{\Gamma\left(b-a\right)}U\left(a,b,z\right)+\frac{e^{\pm(b-a)\pi\mathrm{i}}}{% \Gamma\left(a\right)}e^{z}U\left(b-a,b,e^{\pm\pi\mathrm{i}}z\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : 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 Referenced by: §13.12, §13.14(vii), §13.4(i), §13.7(ii) Permalink: http://dlmf.nist.gov/13.2.E41 Encodings: TeX, pMML, png See also: Annotations for §13.2(vii), §13.2(vii), §13.2 and Ch.13

Also, when $b$ is not an integer

13.2.42		$U\left(a,b,z\right)=\frac{\Gamma\left(1-b\right)}{\Gamma\left(a-b+1\right)}M% \left(a,b,z\right)+\frac{\Gamma\left(b-1\right)}{\Gamma\left(a\right)}z^{1-b}M% \left(a-b+1,2-b,z\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and $z$ : complex variable A&S Ref: 13.1.3 (in different form) Referenced by: §13.14(vii), §13.2(ii), §13.6(v), §13.9(ii), §33.13 Permalink: http://dlmf.nist.gov/13.2.E42 Encodings: TeX, pMML, png See also: Annotations for §13.2(vii), §13.2(vii), §13.2 and Ch.13

13.2.43		$\frac{2\pi\mathrm{i}{\mathrm{e}}^{-z}}{\Gamma\left(b\right)\Gamma\left(a-b+1% \right)}M\left(b-a,b,z\right)={\mathrm{e}}^{b\pi\mathrm{i}}U\left(a,b,{\mathrm% {e}}^{\pi\mathrm{i}}z\right)-{\mathrm{e}}^{-b\pi\mathrm{i}}U\left(a,b,{\mathrm% {e}}^{-\pi\mathrm{i}}z\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : 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 Source: Slater (1960, Eq. (2.2.17)) Referenced by: §13.2(vii), Erratum (V1.2.2) for Additions Permalink: http://dlmf.nist.gov/13.2.E43 Encodings: TeX, pMML, png Addition (effective with 1.2.2): This equation was added. See also: Annotations for §13.2(vii), §13.2(vii), §13.2 and Ch.13

13.2.44		$\frac{2\pi\mathrm{i}{\mathrm{e}}^{-z}}{\Gamma\left(a\right)\Gamma\left(a-b+1% \right)}U\left(b-a,b,z\right)={\mathrm{e}}^{a\pi\mathrm{i}}U\left(a,b,{\mathrm% {e}}^{\pi\mathrm{i}}z\right)-{\mathrm{e}}^{-a\pi\mathrm{i}}U\left(a,b,{\mathrm% {e}}^{-\pi\mathrm{i}}z\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : 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 Source: Slater (1960, Eq. (2.2.18)) Referenced by: §13.2(vii), Erratum (V1.2.2) for Additions Permalink: http://dlmf.nist.gov/13.2.E44 Encodings: TeX, pMML, png Addition (effective with 1.2.2): This equation was added. See also: Annotations for §13.2(vii), §13.2(vii), §13.2 and Ch.13