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21: 7.14 Integrals

… ►

§7.14(i) Error Functions

►

Fourier Transform

… ►

Laplace Transforms

… ►For collections of integrals see Apelblat (1983, pp. 131–146), Erdélyi et al. (1954a, vol. 1, pp. 40, 96, 176–177), Geller and Ng (1971), Gradshteyn and Ryzhik (2000, §§5.4 and 6.28–6.32), Marichev (1983, pp. 184–189), Ng and Geller (1969), Oberhettinger (1974, pp. 138–139, 142–143), Oberhettinger (1990, pp. 48–52, 155–158), Oberhettinger and Badii (1973, pp. 171–172, 179–181), Prudnikov et al. (1986b, vol. 2, pp. 30–36, 93–143), Prudnikov et al. (1992a, §§3.7–3.8), and Prudnikov et al. (1992b, §§3.7–3.8). In a series of ten papers Hadži (1968, 1969, 1970, 1972, 1973, 1975a, 1975b, 1976a, 1976b, 1978) gives many integrals containing error functions and Fresnel integrals, also in combination with the hypergeometric function, confluent hypergeometric functions, and generalized hypergeometric functions.

22: 4.24 Inverse Trigonometric Functions: Further Properties

… ►

4.24.2 $$\arccos z={(2(1-z))}^{1/2}\left(1+\sum _{n=1}^{\mathrm{\infty }}\frac{1\cdot 3\cdot 5\mathrm{\cdots }(2n-1)}{2^{2n}(2n+1)n!}{(1-z)}^n\right),$$ $|1-z|\le 2$.

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\arccos z$: arccosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.4.41 (which is stated differently and the constraint on $z$ is more restrictive.)

Permalink:

http://dlmf.nist.gov/4.24.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(i), §4.24 and Ch.4

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4.24.4 $$\arctan z=\pm \frac{\pi }{2}-\frac{1}{z}+\frac{1}{3z^3}-\frac{1}{5z^5}+\mathrm{\cdots },$$ $\mathrm{\Re }z\gtrless 0$, $|z|\ge 1$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.4.42 (has an error. $\frac{\pi }{2}$ should have a negative sign when .)

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.24.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(i), §4.24 and Ch.4

►

4.24.5 $$\arctan z=\frac{z}{z^2+1}\left(1+\frac{2}{3}\frac{z^2}{1+z^2}+\frac{2\cdot 4}{3\cdot 5}{\left(\frac{z^2}{1+z^2}\right)}^2+\mathrm{\cdots }\right),$$ $\mathrm{\Re }\left(z^2\right)>-\frac{1}{2}$,

ⓘ

Symbols:

$\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.4.42 (has an error in the conditions on $z$.)

Permalink:

http://dlmf.nist.gov/4.24.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(i), §4.24 and Ch.4

… ►

4.24.10 $$\frac{d}{dz}\mathrm{arccsc}z=\mp \frac{1}{z{(z^2-1)}^{1/2}},$$ $\mathrm{\Re }z\gtrless 0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arccsc}z$: arccosecant function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.4.57 (has an error.)

Referenced by:

§4.24(ii)

Permalink:

http://dlmf.nist.gov/4.24.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(ii), §4.24 and Ch.4

►

4.24.11 $$\frac{d}{dz}\mathrm{arcsec}z=\pm \frac{1}{z{(z^2-1)}^{1/2}},$$ $\mathrm{\Re }z\gtrless 0$.

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arcsec}z$: arcsecant function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.4.56 (has an error.)

Referenced by:

§4.24(ii)

Permalink:

http://dlmf.nist.gov/4.24.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(ii), §4.24 and Ch.4

…

23: 7.13 Zeros

… ►

§7.13(i) Zeros of $\mathrm{erf}z$

► $\mathrm{erf}z$ has a simple zero at $z=0$, and in the first quadrant of $ℂ$ there is an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. … ►

§7.13(ii) Zeros of $\mathrm{erfc}z$

►In the sector , $\mathrm{erfc}z$ has an infinite set of zeros $z_n=x_n+\mathrm{i}{y}_n$, $n=1,2,3,\mathrm{\dots }$, arranged in order of increasing absolute value. … ►Thus if $z_n$ is a zero of $\mathrm{erfc}z$ (§7.13(ii)), then $(1+\mathrm{i})z_n/\sqrt{\pi }$ is a zero of $ℱ\left(z\right)$. …

24: 13.18 Relations to Other Functions

… ►

§13.18(ii) Incomplete Gamma Functions

… ►When $\frac{1}{2}-\kappa \pm \mu$ is an integer the Whittaker functions can be expressed as incomplete gamma functions (or generalized exponential integrals). …Special cases are the error functions ►

13.18.6 $$M_{-\frac{1}{4},\frac{1}{4}}\left(z^2\right)=\frac{1}{2}{\mathrm{e}}^{\frac{1}{2}{z}^2}\sqrt{\pi z}\mathrm{erf}\left(z\right),$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(ii), §13.18 and Ch.13

►

13.18.7 $$W_{-\frac{1}{4},\pm \frac{1}{4}}\left(z^2\right)={\mathrm{e}}^{\frac{1}{2}{z}^2}\sqrt{\pi z}\mathrm{erfc}\left(z\right).$$

ⓘ

Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker 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:

Erratum (V1.0.3) for Equation (13.18.7)

Permalink:

http://dlmf.nist.gov/13.18.E7

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.3):

Originally the left-hand side was given correctly as $W_{-\frac{1}{4},-\frac{1}{4}}\left(z^2\right)$; the equation is true also for $W_{-\frac{1}{4},+\frac{1}{4}}\left(z^2\right)$.

See also:

Annotations for §13.18(ii), §13.18 and Ch.13

…

25: 7.8 Inequalities

§7.8 Inequalities

… ►

7.8.2 $x\ge 0$,

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Symbols:

$𝖬\left(x\right)$: Mills’ ratio, $\pi$: the ratio of the circumference of a circle to its diameter and $x$: real variable

A&S Ref:

7.1.13

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E2

Encodings:

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See also:

Annotations for §7.8 and Ch.7

►

7.8.3 $x\ge 0$,

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Symbols:

$𝖬\left(x\right)$: Mills’ ratio, $\pi$: the ratio of the circumference of a circle to its diameter and $x$: real variable

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E3

Encodings:

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See also:

Annotations for §7.8 and Ch.7

… ►The function $F\left(x\right)/\sqrt{1-{\mathrm{e}}^{-2x^2}}$ is strictly decreasing for $x>0$. … ►

7.8.8 $x>0$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm and $x$: real variable

Source:

Pólya (1949, (1.5))

Referenced by:

§7.8, Erratum (V1.0.17) for Equation (7.8.8)

Permalink:

http://dlmf.nist.gov/7.8.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This inequality was added. It is Pólya (1949, (1.5)). Note that it is a special case of (8.10.11).

Suggested by Roberto Iacono

See also:

Annotations for §7.8 and Ch.7

26: 3.2 Linear Algebra

… ►In solving $𝐀𝐱={[1,1,1]}^{\mathrm{T}}$, we obtain by forward elimination $𝐲={[1,-1,3]}^{\mathrm{T}}$, and by back substitution $𝐱={[\frac{1}{6},\frac{1}{6},\frac{1}{6}]}^{\mathrm{T}}$. … ►where $u_j=c_j$, $j=1,2,\mathrm{\dots },n-1$, $d_1=b_1$, and … ►Then we have the a posteriori error bound … ►Start with ${𝐯}_0=\mathrm{𝟎}$, vector ${𝐯}_1$ such that ${𝐯}_1^{\mathrm{T}}𝐒{𝐯}_1=1$, ${\alpha }_1={𝐯}_1^{\mathrm{T}}𝐀{𝐯}_1$, ${\beta }_1=0$. Then for $j=1,2,\mathrm{\dots },n-1$ …

27: 8.4 Special Values

… ►For $\mathrm{erf}\left(z\right)$, $\mathrm{erfc}\left(z\right)$, and $F\left(z\right)$, see §§7.2(i), 7.2(ii). … ►

8.4.1 $$\gamma (\frac{1}{2},z^2)=2{\int }_0^z{\mathrm{e}}^{-t^2}dt=\sqrt{\pi }\mathrm{erf}\left(z\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\int$: integral and $z$: complex variable

A&S Ref:

6.5.16

Permalink:

http://dlmf.nist.gov/8.4.E1

Encodings:

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See also:

Annotations for §8.4 and Ch.8

… ►

8.4.6 $$\mathrm{\Gamma }(\frac{1}{2},z^2)=2{\int }_z^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}dt=\sqrt{\pi }\mathrm{erfc}\left(z\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral and $z$: complex variable

A&S Ref:

6.5.17

Referenced by:

§8.4

Permalink:

http://dlmf.nist.gov/8.4.E6

Encodings:

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See also:

Annotations for §8.4 and Ch.8

►For $n=0,1,2,\mathrm{\dots }$, … ►

8.4.14 $$Q(n+\frac{1}{2},z^2)=\mathrm{erfc}\left(z\right)+\frac{{\mathrm{e}}^{-z^2}}{\sqrt{\pi }}\sum _{k=1}^n\frac{z^{2k-1}}{{\left(\frac{1}{2}\right)}_k},$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§8.4

Permalink:

http://dlmf.nist.gov/8.4.E14

Encodings:

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See also:

Annotations for §8.4 and Ch.8

…

28: 5.19 Mathematical Applications

… ►

$a_k={\displaystyle \frac{k}{(3k+2)(2k+1)(k+1)}}.$

… ►

5.19.2 $$a_k=\frac{2}{k+\frac{2}{3}}-\frac{1}{k+\frac{1}{2}}-\frac{1}{k+1}=\left(\frac{1}{k+1}-\frac{1}{k+\frac{1}{2}}\right)-2\left(\frac{1}{k+1}-\frac{1}{k+\frac{2}{3}}\right).$$

ⓘ

Symbols:

$k$: nonnegative integer and $a$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.19(i), §5.19(i), §5.19 and Ch.5

… ►By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of $f(z)$ for large $|z|$, or small $|z|$, can be obtained complete with an integral representation of the error term. … ►

$V={\displaystyle \frac{{\pi }^{\frac{1}{2}n}{r}^n}{\mathrm{\Gamma }\left(\frac{1}{2}n+1\right)}},$

►

$S={\displaystyle \frac{2{\pi }^{\frac{1}{2}n}{r}^{n-1}}{\mathrm{\Gamma }\left(\frac{1}{2}n\right)}}={\displaystyle \frac{n}{r}}V.$

29: 7.7 Integral Representations

… ►

§7.7(i) Error Functions and Dawson’s Integral

►Integrals of the type $\int {\mathrm{e}}^{-z^2}R(z)dz$, where $R(z)$ is an arbitrary rational function, can be written in closed form in terms of the error functions and elementary functions. … ►

7.7.9 $${\int }_0^x\mathrm{erf}tdt=x\mathrm{erf}x+\frac{1}{\sqrt{\pi }}\left({\mathrm{e}}^{-x^2}-1\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable

A&S Ref:

7.4.35 (in different form)

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

… ►In (7.7.13) and (7.7.14) the integration paths are straight lines, $\zeta =\frac{1}{16}{\pi }^2{z}^4$, and $c$ is a constant such that in (7.7.13), and in (7.7.14). … ►For other integral representations see Erdélyi et al. (1954a, vol. 1, pp. 265–267, 270), Ng and Geller (1969), Oberhettinger (1974, pp. 246–248), and Oberhettinger and Badii (1973, pp. 371–377).

30: 32.10 Special Function Solutions

… ►with $C_1$, $C_2$ arbitrary constants. … ►More generally, if $n=1,2,3,\mathrm{\dots }$, then … ►If $n=1$, then the Riccati equation is …with $C_1$, $C_2$ arbitrary constants. … ►with $C_1$, $C_2$ arbitrary constants. …