# §13.2 Definitions and Basic Properties

## §13.2(i) Differential Equation

### Kummer’s Equation

 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

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

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 $\lim_{b\to-n}\frac{M\left(a,b,z\right)}{\Gamma\left(b\right)}={\mathbf{M}}% \left(a,-n,z\right)=\frac{{\left(a\right)_{n+1}}}{(n+1)!}z^{n+1}M\left(a+n+1,n% +2,z\right).$

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), item 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), item 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, 5th item 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), item Equation (13.2.10), 5th item 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

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

 13.2.12 $U\left(a,b,ze^{2\pi\mathrm{i}m}\right)=\frac{2\pi\mathrm{i}e^{-\pi\mathrm{i}bm% }\sin\left(\pi bm\right)}{\Gamma\left(1+a-b\right)\sin\left(\pi b\right)}{% \mathbf{M}}\left(a,b,z\right)+e^{-2\pi\mathrm{i}bm}U\left(a,b,z\right).$

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 $z\to 0$

 13.2.13 $M\left(a,b,z\right)=1+O\left(z\right).$

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

 13.2.14 $U\left(-n,b,z\right)=(-1)^{n}{\left(b\right)_{n}}+O\left(z\right),$
 13.2.15 $U\left(-n+b-1,b,z\right)=(-1)^{n}{\left(2-b\right)_{n}}z^{1-b}+O\left(z^{2-b}% \right).$

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 $\displaystyle U\left(a,2,z\right)$ $\displaystyle=\frac{1}{\Gamma\left(a\right)}z^{-1}+O\left(\ln z\right),$ 13.2.18 $\displaystyle U\left(a,b,z\right)$ $\displaystyle=\frac{\Gamma\left(b-1\right)}{\Gamma\left(a\right)}z^{1-b}+\frac% {\Gamma\left(1-b\right)}{\Gamma\left(a-b+1\right)}+O\left(z^{2-\Re b}\right),$ $1\leq\Re b<2$, $b\not=1$, 13.2.19 $\displaystyle U\left(a,1,z\right)$ $\displaystyle=-\frac{1}{\Gamma\left(a\right)}\left(\ln z+\psi\left(a\right)+2% \gamma\right)+O\left(z\ln z\right),$ 13.2.20 $\displaystyle U\left(a,b,z\right)$ $\displaystyle=\frac{\Gamma\left(1-b\right)}{\Gamma\left(a-b+1\right)}+O\left(z% ^{1-\Re b}\right),$ $0<\Re b<1$, 13.2.21 $\displaystyle U\left(a,0,z\right)$ $\displaystyle=\frac{1}{\Gamma\left(a+1\right)}+O\left(z\ln z\right),$ 13.2.22 $\displaystyle U\left(a,b,z\right)$ $\displaystyle=\frac{\Gamma\left(1-b\right)}{\Gamma\left(a-b+1\right)}+O\left(z% \right),$ $\Re b\leq 0$, $b\not=0$.

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

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

 13.2.23 ${\mathbf{M}}\left(a,b,z\right)\sim\ifrac{e^{z}z^{a-b}}{\Gamma\left(a\right)},$ $\left|\operatorname{ph}z\right|\leq\frac{1}{2}\pi-\delta$,

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

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

 13.2.24 $U\left(a,b,z\right)$, $e^{z}U\left(b-a,b,e^{-\pi\mathrm{i}}z\right)$, $-\tfrac{1}{2}\pi\leq\operatorname{ph}{z}\leq\tfrac{3}{2}\pi$, 13.2.25 $U\left(a,b,z\right)$, $e^{z}U\left(b-a,b,e^{\pi\mathrm{i}}z\right)$, $-\tfrac{3}{2}\pi\leq\operatorname{ph}{z}\leq\tfrac{1}{2}\pi$.

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

 13.2.26 $M\left(a,b,z\right),\quad z^{1-b}M\left(a-b+1,2-b,z\right),$ $b\not\in\mathbb{Z}$.

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 $\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(1-a\right)_{k}}}z^{-k}-\sum_{k=0}^{% \infty}\frac{{\left(a\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\ln z+% \psi\left(a+k\right)-\psi\left(1+k\right)-\psi\left(n+k+1\right)\right),$

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

 13.2.28 $\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(1-a\right)_{k}}}z^{-k}-\sum_{k=0}^{% -a}\frac{{\left(a\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^{k}\left(\ln z+\psi% \left(1-a-k\right)-\psi\left(1+k\right)-\psi\left(n+k+1\right)\right)+(-1)^{1-% a}(-a)!\sum_{k=1-a}^{\infty}\frac{(k-1+a)!}{{\left(n+1\right)_{k}}k!}z^{k},$

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 $\sum_{k=1}^{n+1}\frac{(n+1)!(k-1)!}{(n-k+1)!{\left(-a-n\right)_{k}}}z^{n-k+1}-% \sum_{k=0}^{\infty}\frac{{\left(a+n+1\right)_{k}}}{{\left(n+2\right)_{k}}k!}z^% {n+k+1}\left(\ln z+\psi\left(a+n+k+1\right)-\psi\left(1+k\right)-\psi\left(n+k% +2\right)\right),$

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

 13.2.31 $\sum_{k=1}^{n+1}\frac{(n+1)!(k-1)!}{(n-k+1)!{\left(-a-n\right)_{k}}}z^{n-k+1}-% \sum_{k=0}^{-a-n-1}\frac{{\left(a+n+1\right)_{k}}}{{\left(n+2\right)_{k}}k!}z^% {n+k+1}\left(\ln z+\psi\left(-a-n-k\right)-\psi\left(1+k\right)-\psi\left(n+k+% 2\right)\right)+(-1)^{n-a}{(-a-n-1)!}\sum_{k=-a-n}^{\infty}\frac{(k+a+n)!}{{% \left(n+2\right)_{k}}k!}z^{n+k+1},$

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

 13.2.33 $\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),z^{1-b}{\mathbf{% M}}\left(a-b+1,2-b,z\right)\right\}$ $\displaystyle=\sin\left(\pi b\right)z^{-b}e^{z}/\pi,$ 13.2.34 $\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),U\left(a,b,z% \right)\right\}$ $\displaystyle=-\ifrac{z^{-b}e^{z}}{\Gamma\left(a\right)},$ 13.2.35 $\displaystyle\mathscr{W}\left\{{\mathbf{M}}\left(a,b,z\right),e^{z}U\left(b-a,% b,e^{\pm\pi\mathrm{i}}z\right)\right\}$ $\displaystyle=\ifrac{e^{\mp b\pi\mathrm{i}}z^{-b}e^{z}}{\Gamma\left(b-a\right)},$ 13.2.36 $\displaystyle\mathscr{W}\left\{z^{1-b}{\mathbf{M}}\left(a-b+1,2-b,z\right),U% \left(a,b,z\right)\right\}$ $\displaystyle=-\ifrac{z^{-b}e^{z}}{\Gamma\left(a-b+1\right)},$ 13.2.37 $\displaystyle\mathscr{W}\left\{z^{1-b}{\mathbf{M}}\left(a-b+1,2-b,z\right),e^{% z}U\left(b-a,b,e^{\pm\pi\mathrm{i}}z\right)\right\}$ $\displaystyle=-\ifrac{z^{-b}e^{z}}{\Gamma\left(1-a\right)},$ 13.2.38 $\displaystyle\mathscr{W}\left\{U\left(a,b,z\right),e^{z}U\left(b-a,b,e^{\pm\pi% \mathrm{i}}z\right)\right\}$ $\displaystyle=e^{\pm(a-b)\pi\mathrm{i}}z^{-b}e^{z}.$

## §13.2(vii) Connection Formulas

### Kummer’s Transformations

 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 $\frac{1}{\Gamma\left(b\right)}M\left(a,b,z\right)=\frac{e^{\mp a\pi\mathrm{i}}% }{\Gamma\left(b-a\right)}U\left(a,b,z\right)+\frac{e^{\pm(b-a)\pi\mathrm{i}}}{% \Gamma\left(a\right)}e^{z}U\left(b-a,b,e^{\pm\pi\mathrm{i}}z\right).$

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