§13.6 Relations to Other Functions

13.6.1		$M(a,a,z)$	$={\mathrm{e}}^z,$
ⓘ Symbols: $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable A&S Ref: 13.6.12 Referenced by: §13.10(v) Permalink: http://dlmf.nist.gov/13.6.E1 Encodings: TeX, pMML, png See also: Annotations for §13.6(i), §13.6 and Ch.13
13.6.2		$M(1,2,2z)$	$={\displaystyle \frac{{\mathrm{e}}^z}{z}}\sinh z,$
ⓘ Symbols: $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function and $z$: complex variable A&S Ref: 13.6.14 Permalink: http://dlmf.nist.gov/13.6.E2 Encodings: TeX, pMML, png See also: Annotations for §13.6(i), §13.6 and Ch.13
13.6.3		$M(0,b,z)$	$=U(0,b,z)=1,$
ⓘ 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: TeX, pMML, png See also: Annotations for §13.6(i), §13.6 and Ch.13
13.6.4		$U(a,a+1,z)$	$=z^{-a}.$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable Permalink: http://dlmf.nist.gov/13.6.E4 Encodings: TeX, pMML, png See also: Annotations for §13.6(i), §13.6 and Ch.13

For the notation see §§6.2(i), 7.2(i), 8.2(i), and 8.19(i). 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). For example,

13.6.5		$$M(a,a+1,-z)={\mathrm{e}}^{-z}M(1,a+1,z)=a{z}^{-a}\gamma (a,z),$$
ⓘ Symbols: $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function and $z$: complex variable A&S Ref: 13.6.10 Permalink: http://dlmf.nist.gov/13.6.E5 Encodings: TeX, pMML, png See also: Annotations for §13.6(ii), §13.6 and Ch.13

13.6.6		$$U(a,a,z)=z^{1-a}U(1,2-a,z)=z^{1-a}{\mathrm{e}}^z{E}_a\left(z\right)={\mathrm{e}}^z\mathrm{\Gamma }(1-a,z).$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable A&S Ref: 13.6.28 (in different form) Permalink: http://dlmf.nist.gov/13.6.E6 Encodings: TeX, pMML, png See also: Annotations for §13.6(ii), §13.6 and Ch.13

Special cases are the error functions

13.6.7		$$M(\frac{1}{2},\frac{3}{2},-z^2)=\frac{\sqrt{\pi }}{2z}\mathrm{erf}\left(z\right),$$
ⓘ 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 and $z$: complex variable A&S Ref: 13.6.19 Referenced by: §13.6(ii) Permalink: http://dlmf.nist.gov/13.6.E7 Encodings: TeX, pMML, png See also: Annotations for §13.6(ii), §13.6 and Ch.13

13.6.8		$$U(\frac{1}{2},\frac{1}{2},z^2)=\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}\left(z\right).$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable Referenced by: §13.6(ii) Permalink: http://dlmf.nist.gov/13.6.E8 Encodings: TeX, pMML, png See also: Annotations for §13.6(ii), §13.6 and Ch.13

When $b=2a$ the Kummer functions can be expressed as modified Bessel functions. For the notation see §§10.25(ii) and 9.2(i).

13.6.9		$M(\nu +\frac{1}{2},2\nu +1,2z)$	$=\mathrm{\Gamma }\left(1+\nu \right){\mathrm{e}}^z{\left(z/2\right)}^{-\nu }{I}_{\nu }\left(z\right),$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $z$: complex variable A&S Ref: 13.6.3 Referenced by: §13.11, §7.6(ii) Permalink: http://dlmf.nist.gov/13.6.E9 Encodings: TeX, pMML, png See also: Annotations for §13.6(iii), §13.6 and Ch.13
13.6.10		$U(\nu +\frac{1}{2},2\nu +1,2z)$	$={\displaystyle \frac{1}{\sqrt{\pi }}}{\mathrm{e}}^z{\left(2z\right)}^{-\nu }{K}_{\nu }\left(z\right),$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $z$: complex variable A&S Ref: 13.6.21 Permalink: http://dlmf.nist.gov/13.6.E10 Encodings: TeX, pMML, png See also: Annotations for §13.6(iii), §13.6 and Ch.13
13.6.11		$U(\frac{5}{6},\frac{5}{3},\frac{4}{3}{z}^{3/2})$	$=\sqrt{\pi }{\displaystyle \frac{3^{5/6}\exp \left(\frac{2}{3}{z}^{3/2}\right)}{2^{2/3}z}}\mathrm{Ai}\left(z\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function and $z$: complex variable A&S Ref: 13.6.25 Referenced by: (9.10.18) Permalink: http://dlmf.nist.gov/13.6.E11 Encodings: TeX, pMML, png See also: Annotations for §13.6(iii), §13.6 and Ch.13

and in the case that $b-2a$ is an integer we have

13.6.11_1		$M(\nu +\frac{1}{2},2\nu +1+n,2z)$	$=\mathrm{\Gamma }\left(\nu \right){\mathrm{e}}^z{\left(z/2\right)}^{-\nu }{\displaystyle \sum _{k=0}^n}{\displaystyle \frac{{\left(-n\right)}_k{\left(2\nu \right)}_k(\nu +k)}{{\left(2\nu +1+n\right)}_{k}k!}}{I}_{\nu +k}\left(z\right),$
ⓘ 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: TeX, pMML, png Addition (effective with 1.1.0): This equation was added. See also: Annotations for §13.6(iii), §13.6 and Ch.13
13.6.11_2		$M(\nu +\frac{1}{2},2\nu +1-n,2z)$	$=\mathrm{\Gamma }\left(\nu -n\right){\mathrm{e}}^z{\left(z/2\right)}^{n-\nu }{\displaystyle \sum _{k=0}^n}{(-1)}^k{\displaystyle \frac{{\left(-n\right)}_k{\left(2\nu -2n\right)}_k(\nu -n+k)}{{\left(2\nu +1-n\right)}_{k}k!}}{I}_{\nu +k-n}\left(z\right).$
ⓘ 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, (7.11.1.7)) Referenced by: §13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/13.6.E11_2 Encodings: TeX, pMML, png Addition (effective with 1.1.0): This equation was added. See also: Annotations for §13.6(iii), §13.6 and Ch.13

Note that (13.6.11_1) and (13.6.11_2) are special cases of (13.11.1) and (13.11.2), respectively

For the notation see §12.2.

13.6.12		$$U(\frac{1}{2}a+\frac{1}{4},\frac{1}{2},\frac{1}{2}{z}^2)=2^{\frac{1}{2}a+\frac{1}{4}}{\mathrm{e}}^{\frac{1}{4}{z}^2}U(a,z),$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable Permalink: http://dlmf.nist.gov/13.6.E12 Encodings: TeX, pMML, png See also: Annotations for §13.6(iv), §13.6 and Ch.13

13.6.13		$$U(\frac{1}{2}a+\frac{3}{4},\frac{3}{2},\frac{1}{2}{z}^2)=2^{\frac{1}{2}a+\frac{3}{4}}\frac{{\mathrm{e}}^{\frac{1}{4}{z}^2}}{z}U(a,z).$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable Permalink: http://dlmf.nist.gov/13.6.E13 Encodings: TeX, pMML, png See also: Annotations for §13.6(iv), §13.6 and Ch.13

13.6.14		$$M(\frac{1}{2}a+\frac{1}{4},\frac{1}{2},\frac{1}{2}{z}^2)=\frac{2^{\frac{1}{2}a-\frac{3}{4}}\mathrm{\Gamma }\left(\frac{1}{2}a+\frac{3}{4}\right){\mathrm{e}}^{\frac{1}{4}{z}^2}}{\sqrt{\pi }}\left(U(a,z)+U(a,-z)\right),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable Permalink: http://dlmf.nist.gov/13.6.E14 Encodings: TeX, pMML, png See also: Annotations for §13.6(iv), §13.6 and Ch.13

13.6.15		$$M(\frac{1}{2}a+\frac{3}{4},\frac{3}{2},\frac{1}{2}{z}^2)=\frac{2^{\frac{1}{2}a-\frac{5}{4}}\mathrm{\Gamma }\left(\frac{1}{2}a+\frac{1}{4}\right){\mathrm{e}}^{\frac{1}{4}{z}^2}}{z\sqrt{\pi }}\left(U(a,-z)-U(a,z)\right).$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_{1}F_1(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable Permalink: http://dlmf.nist.gov/13.6.E15 Encodings: TeX, pMML, png See also: Annotations for §13.6(iv), §13.6 and Ch.13

Special cases of §13.6(iv) are as follows. For the notation see §§18.3, 18.19.

13.6.21		$$U(a,b,z)=z^{-a}{}_{2}F_0(a,a-b+1;-;-z^{-1}).$$
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$ generalized hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable Referenced by: §13.6(vi) Permalink: http://dlmf.nist.gov/13.6.E21 Encodings: TeX, pMML, png See also: Annotations for §13.6(vi), §13.6 and Ch.13

For the definition of ${}_{2}F_0(a,a-b+1;-;-z^{-1})$ when neither $a$ nor $a-b+1$ is a nonpositive integer see §16.5.

For representations of Coulomb functions in terms of Kummer functions see (33.2.4), (33.2.8) and (33.14.5).