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31: 13.6 Relations to Other Functions

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13.6.1 $$M(a,a,z)={\mathrm{e}}^z,$$

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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:

pMML, png, TeX

See also:

Annotations for §13.6(i), §13.6 and Ch.13

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13.6.2 $$M(1,2,2z)=\frac{{\mathrm{e}}^z}{z}\sinh z,$$

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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:

pMML, png, TeX

See also:

Annotations for §13.6(i), §13.6 and Ch.13

… ►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). … ►

13.6.5 $$M(a,a+1,-z)={\mathrm{e}}^{-z}M(1,a+1,z)=a{z}^{-a}\gamma (a,z),$$

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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:

pMML, png, TeX

See also:

Annotations for §13.6(ii), §13.6 and Ch.13

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13.6.8 $$U(\frac{1}{2},\frac{1}{2},z^2)=\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}\left(z\right).$$

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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:

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See also:

Annotations for §13.6(ii), §13.6 and Ch.13

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32: 20.6 Power Series

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20.6.2 $${\theta }_1\left(\pi z\right|\tau )=\pi z{\theta }_1^{\prime }\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\delta }_{2j}(\tau )z^{2j}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\delta }_{2j}(\tau )$: coefficient

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.6.E2

Encodings:

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See also:

Annotations for §20.6 and Ch.20

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20.6.3 $${\theta }_2\left(\pi z\right|\tau )={\theta }_2\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\alpha }_{2j}(\tau )z^{2j}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\alpha }_{2j}(\tau )$: coefficient

Permalink:

http://dlmf.nist.gov/20.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

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20.6.4 $${\theta }_3\left(\pi z\right|\tau )={\theta }_3\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\beta }_{2j}(\tau )z^{2j}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\beta }_{2j}(\tau )$: coefficient

Permalink:

http://dlmf.nist.gov/20.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

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20.6.5 $${\theta }_4\left(\pi z\right|\tau )={\theta }_4\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\gamma }_{2j}(\tau )z^{2j}\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\gamma }_{2j}(\tau )$: coefficient

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

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33: 8.8 Recurrence Relations and Derivatives

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8.8.4 $$z{\gamma }^{\ast }(a+1,z)={\gamma }^{\ast }(a,z)-\frac{{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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8.8.5 $$P(a+1,z)=P(a,z)-\frac{z^a{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.21

Referenced by:

§8.25(v)

Permalink:

http://dlmf.nist.gov/8.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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8.8.6 $$Q(a+1,z)=Q(a,z)+\frac{z^a{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a+1\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.25(v), §8.4

Permalink:

http://dlmf.nist.gov/8.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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8.8.11 $$P(a+n,z)=P(a,z)-z^a{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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8.8.12 $$Q(a+n,z)=Q(a,z)+z^a{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §8.8 and Ch.8

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34: 32.15 Orthogonal Polynomials

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32.15.1 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(-\frac{1}{4}{\xi }^4-z{\xi }^2\right)p_m(\xi )p_n(\xi )d\xi ={\delta }_{m,n},$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $m$: integer, $n$: integer, $z$: real and $p_n(\xi )$: polynomials

Permalink:

http://dlmf.nist.gov/32.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.15 and Ch.32

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35: 10.14 Inequalities; Monotonicity

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10.14.4 $$|J_{\nu }\left(z\right)|\le \frac{{|\frac{1}{2}z|}^{\nu }{\mathrm{e}}^{|\mathrm{\Im }z|}}{\mathrm{\Gamma }\left(\nu +1\right)},$$ $\nu \ge -\frac{1}{2}$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.62 (corrected)

Permalink:

http://dlmf.nist.gov/10.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.14 and Ch.10

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10.14.5 $$|J_{\nu }\left(\nu x\right)|\le \frac{x^{\nu }\exp \left(\nu {(1-x^2)}^{\frac{1}{2}}\right)}{{\left(1+{(1-x^2)}^{\frac{1}{2}}\right)}^{\nu }},$$ ;

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\exp z$: exponential function, $x$: real variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.14 and Ch.10

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10.14.6 $$|J_{\nu }^{\prime }\left(\nu x\right)|\le \frac{{(1+x^2)}^{\frac{1}{4}}}{x{(2\pi \nu )}^{\frac{1}{2}}}\frac{x^{\nu }\exp \left(\nu {(1-x^2)}^{\frac{1}{2}}\right)}{{\left(1+{(1-x^2)}^{\frac{1}{2}}\right)}^{\nu }},$$ ;

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $x$: real variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.14 and Ch.10

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10.14.7 $$1\le \frac{J_{\nu }\left(\nu x\right)}{x^{\nu }{J}_{\nu }\left(\nu \right)}\le {\mathrm{e}}^{\nu (1-x)},$$ ;

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{e}$: base of natural logarithm, $x$: real variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.14 and Ch.10

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10.14.8 $$|J_n\left(nz\right)|\le \frac{\left|z^n\exp \left(n{(1-z^2)}^{\frac{1}{2}}\right)\right|}{{\left|1+{(1-z^2)}^{\frac{1}{2}}\right|}^n},$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\exp z$: exponential function, $n$: integer and $z$: complex variable

A&S Ref:

9.1.63

Permalink:

http://dlmf.nist.gov/10.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.14, §10.14 and Ch.10

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36: 4.5 Inequalities

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§4.5(ii) Exponentials

… ►For more inequalities involving the exponential function see Mitrinović (1964, pp. 73–77), Mitrinović (1970, pp. 266–271), and Bullen (1998, pp. 81–83).

37: 8.7 Series Expansions

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8.7.1 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)}=\frac{1}{\mathrm{\Gamma }\left(a\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-z)}^k}{k!(a+k)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

A&S Ref:

6.5.29

Referenced by:

§8.11(ii), §8.14, §8.6(i), §8.7

Permalink:

http://dlmf.nist.gov/8.7.E1

Encodings:

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See also:

Annotations for §8.7 and Ch.8

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8.7.2 $$\gamma (a,x+y)-\gamma (a,x)=\mathrm{\Gamma }(a,x)-\mathrm{\Gamma }(a,x+y)={\mathrm{e}}^{-x}{x}^{a-1}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_n}{{(-x)}^n}(1-{\mathrm{e}}^{-y}{e}_n(y)),$$ .

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\gamma (a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter, $n$: nonnegative integer and $e_n(z)$: functions

A&S Ref:

6.5.30

Permalink:

http://dlmf.nist.gov/8.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

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8.7.3 $$\mathrm{\Gamma }(a,z)=\mathrm{\Gamma }\left(a\right)-\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{z}^{a+k}}{k!(a+k)}=\mathrm{\Gamma }\left(a\right)\left(1-z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{\mathrm{\Gamma }\left(a+k+1\right)}\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Referenced by:

§8.14, §8.19(iv), §8.4, §8.7

Permalink:

http://dlmf.nist.gov/8.7.E3

Encodings:

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See also:

Annotations for §8.7 and Ch.8

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8.7.4 $$\gamma (a,x)=\mathrm{\Gamma }\left(a\right)x^{\frac{1}{2}a}{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}{e}_n(-1)x^{\frac{1}{2}n}{I}_{n+a}\left(2x^{1/2}\right),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $a$: parameter, $n$: nonnegative integer and $e_n(z)$: functions

Permalink:

http://dlmf.nist.gov/8.7.E4

Encodings:

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See also:

Annotations for §8.7 and Ch.8

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8.7.5 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-\frac{1}{2}z}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_n}{\mathrm{\Gamma }\left(n+a+1\right)}\left(2n+1\right){𝗂}_n^{(1)}\left(\frac{1}{2}z\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Referenced by:

§8.7

Permalink:

http://dlmf.nist.gov/8.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

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38: 7.10 Derivatives

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7.10.1 $$\frac{d^{n+1}\mathrm{erf}z}{{dz}^{n+1}}={(-1)}^n\frac{2}{\sqrt{\pi }}{H}_n\left(z\right){\mathrm{e}}^{-z^2},$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.19

Permalink:

http://dlmf.nist.gov/7.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

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39: 6.13 Zeros

… ►The function $\mathrm{Ei}\left(x\right)$ has one real zero $x_0$, given by …

40: 4.28 Definitions and Periodicity

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4.28.1 $$\sinh z=\frac{{\mathrm{e}}^z-{\mathrm{e}}^{-z}}{2},$$

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Defines:

$\sinh z$: hyperbolic sine function

Symbols:

$\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

4.5.1

Referenced by:

§4.45(i), §4.45(ii)

Permalink:

http://dlmf.nist.gov/4.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.28 and Ch.4

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4.28.2 $$\cosh z=\frac{{\mathrm{e}}^z+{\mathrm{e}}^{-z}}{2},$$

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Defines:

$\cosh z$: hyperbolic cosine function

Symbols:

$\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

4.5.2

Permalink:

http://dlmf.nist.gov/4.28.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.28 and Ch.4

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4.28.3 $$\cosh z\pm \sinh z={\mathrm{e}}^{\pm z},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

4.5.19 4.5.20

Permalink:

http://dlmf.nist.gov/4.28.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.28 and Ch.4

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