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31: 7.1 Special Notation

… ► ►►►

$x$	real variable.
$z$	complex variable.
…

… ►The main functions treated in this chapter are the error function $\mathrm{erf}z$; the complementary error functions $\mathrm{erfc}z$ and $w\left(z\right)$; Dawson’s integral $F\left(z\right)$; the Fresnel integrals $ℱ\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$; the Goodwin–Staton integral $G\left(z\right)$; the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$; the Voigt functions $𝖴(x,t)$ and $𝖵(x,t)$. ►Alternative notations are $Q(z)=\frac{1}{2}\mathrm{erfc}\left(z/\sqrt{2}\right)$, $P(z)=\mathrm{\Phi }(z)=\frac{1}{2}\mathrm{erfc}\left(-z/\sqrt{2}\right)$, $\mathrm{Erf}z=\frac{1}{2}\sqrt{\pi }\mathrm{erf}z$, $\mathrm{Erfi}z={\mathrm{e}}^{z^2}F\left(z\right)$, $C_1(z)=C\left(\sqrt{2/\pi }z\right)$, $S_1(z)=S\left(\sqrt{2/\pi }z\right)$, $C_2(z)=C\left(\sqrt{2z/\pi }\right)$, $S_2(z)=S\left(\sqrt{2z/\pi }\right)$. ►The notations $P(z)$, $Q(z)$, and $\mathrm{\Phi }(z)$ are used in mathematical statistics, where these functions are called the normal or Gaussian probability functions. …

32: 13.3 Recurrence Relations and Derivatives

… ►

13.3.7 $$U(a-1,b,z)+(b-2a-z)U(a,b,z)+a(a-b+1)U(a+1,b,z)=0,$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.15

Permalink:

http://dlmf.nist.gov/13.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

►

13.3.8 $$(b-a-1)U(a,b-1,z)+(1-b-z)U(a,b,z)+zU(a,b+1,z)=0,$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.16

Permalink:

http://dlmf.nist.gov/13.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

… ►

13.3.10 $$(b-a)U(a,b,z)+U(a-1,b,z)-zU(a,b+1,z)=0,$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.18

Permalink:

http://dlmf.nist.gov/13.3.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(i), §13.3 and Ch.13

►

13.3.11 $$(a+z)U(a,b,z)-zU(a,b+1,z)+a(b-a-1)U(a+1,b,z)=0,$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.19

Permalink:

http://dlmf.nist.gov/13.3.E11

Encodings:

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See also:

Annotations for §13.3(i), §13.3 and Ch.13

►

13.3.12 $$(a-1+z)U(a,b,z)-U(a-1,b,z)+(a-b+1)U(a,b-1,z)=0.$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function and $z$: complex variable

A&S Ref:

13.4.20

Permalink:

http://dlmf.nist.gov/13.3.E12

Encodings:

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See also:

Annotations for §13.3(i), §13.3 and Ch.13

…

33: 15.5 Derivatives and Contiguous Functions

… ►

15.5.5 $${\left(z\frac{d}{dz}z\right)}^n\left(z^{c-a-1}{(1-z)}^{a+b-c}F(a,b;c;z)\right)={\left(c-a\right)}_n{z}^{c-a+n-1}{(1-z)}^{a-n+b-c}F(a-n,b;c;z).$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§18.17(iv)

Permalink:

http://dlmf.nist.gov/15.5.E5

Encodings:

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See also:

Annotations for §15.5(i), §15.5 and Ch.15

… ►

15.5.10 $${\left(z\frac{d}{dz}z\right)}^n=z^n\frac{d^n}{{dz}^n}{z}^n,$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $n$: integer and $z$: complex variable

Referenced by:

§15.5(i)

Permalink:

http://dlmf.nist.gov/15.5.E10

Encodings:

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See also:

Annotations for §15.5(i), §15.5 and Ch.15

… ►The six functions $F(a\pm 1,b;c;z)$, $F(a,b\pm 1;c;z)$, $F(a,b;c\pm 1;z)$ are said to be contiguous to $F(a,b;c;z)$. … ►By repeated applications of (15.5.11)–(15.5.18) any function $F(a+k,b+\mathrm{\ell };c+m;z)$, in which $k,\mathrm{\ell },m$ are integers, can be expressed as a linear combination of $F(a,b;c;z)$ and any one of its contiguous functions, with coefficients that are rational functions of $a,b,c$, and $z$. … ►

15.5.20 $$z(1-z)\left(dF(a,b;c;z)/dz\right)=(c-a)F(a-1,b;c;z)+(a-c+bz)F(a,b;c;z)=(c-b)F(a,b-1;c;z)+(b-c+az)F(a,b;c;z),$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.5.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §15.5(ii), §15.5 and Ch.15

…

34: 4.22 Infinite Products and Partial Fractions

… ►

4.22.1 $$\sin z=z\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z^2}{n^2{\pi }^2}\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.89

Permalink:

http://dlmf.nist.gov/4.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

►

4.22.2 $$\cos z=\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{4z^2}{{(2n-1)}^2{\pi }^2}\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.90

Permalink:

http://dlmf.nist.gov/4.22.E2

Encodings:

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See also:

Annotations for §4.22 and Ch.4

►When $z\ne n\pi$, $n\in \mathbb{Z}$, ►

4.22.3 $$\cot z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{1}{z^2-n^2{\pi }^2},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.91

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/4.22.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.22 and Ch.4

… ►

4.22.5 $$\csc z=\frac{1}{z}+2z\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{z^2-n^2{\pi }^2}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.93

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/4.22.E5

Encodings:

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See also:

Annotations for §4.22 and Ch.4

35: 4.39 Continued Fractions

… ►

4.39.1 $$\tanh z=\frac{z}{1+}\frac{z^2}{3+}\frac{z^2}{5+}\frac{z^2}{7+}\mathrm{\cdots },$$ $z\ne \pm \frac{1}{2}\pi \mathrm{i},\pm \frac{3}{2}\pi \mathrm{i},\mathrm{\dots }$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.5.70

Permalink:

http://dlmf.nist.gov/4.39.E1

Encodings:

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See also:

Annotations for §4.39 and Ch.4

►

4.39.2 $$\frac{\mathrm{arcsinh}z}{\sqrt{1+z^2}}=\frac{z}{1+}\frac{1\cdot 2z^2}{3+}\frac{1\cdot 2z^2}{5+}\frac{3\cdot 4z^2}{7+}\frac{3\cdot 4z^2}{9+}\mathrm{\cdots },$$

ⓘ

Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function and $z$: complex variable

A&S Ref:

4.6.36

Permalink:

http://dlmf.nist.gov/4.39.E2

Encodings:

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Annotations for §4.39 and Ch.4

►where $z$ is in the open cut plane of Figure 4.37.1(i). ►

4.39.3 $$\mathrm{arctanh}z=\frac{z}{1-}\frac{z^2}{3-}\frac{4z^2}{5-}\frac{9z^2}{7-}\mathrm{\cdots },$$

ⓘ

Symbols:

$\mathrm{arctanh}z$: inverse hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.6.35

Permalink:

http://dlmf.nist.gov/4.39.E3

Encodings:

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Annotations for §4.39 and Ch.4

►where $z$ is in the open cut plane of Figure 4.37.1(iii). …

36: 8.5 Confluent Hypergeometric Representations

… ►

8.5.1 $$\gamma (a,z)=a^{-1}{z}^a{\mathrm{e}}^{-z}M(1,1+a,z)=a^{-1}{z}^{a}M(a,1+a,-z),$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

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, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.12

Referenced by:

§8.11(ii), §8.14

Permalink:

http://dlmf.nist.gov/8.5.E1

Encodings:

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See also:

Annotations for §8.5 and Ch.8

►

8.5.2 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-z}𝐌(1,1+a,z)=𝐌(a,1+a,-z).$$

ⓘ

Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.7

Permalink:

http://dlmf.nist.gov/8.5.E2

Encodings:

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See also:

Annotations for §8.5 and Ch.8

►

8.5.3 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-z}U(1-a,1-a,z)=z^a{\mathrm{e}}^{-z}U(1,1+a,z).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.14, §8.19(vi)

Permalink:

http://dlmf.nist.gov/8.5.E3

Encodings:

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See also:

Annotations for §8.5 and Ch.8

►

8.5.4 $$\gamma (a,z)=a^{-1}{z}^{\frac{1}{2}a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}z}{M}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.5

Permalink:

http://dlmf.nist.gov/8.5.E4

Encodings:

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See also:

Annotations for §8.5 and Ch.8

►

8.5.5 $$\mathrm{\Gamma }(a,z)={\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}a-\frac{1}{2}}{W}_{\frac{1}{2}a-\frac{1}{2},\frac{1}{2}a}\left(z\right).$$

ⓘ

Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

37: 4.8 Identities

… ►In (4.8.1)–(4.8.4) $z_1{z}_2\ne 0$. …This is interpreted that every value of $\mathrm{Ln}\left(z_1{z}_2\right)$ is one of the values of $\mathrm{Ln}{z}_1+\mathrm{Ln}{z}_2$, and vice versa. … ►In (4.8.5)–(4.8.7) and (4.8.10) $z\ne 0$. … ►If $a\ne 0$ and $a^z$ has its general value, then … ►The restriction on $z_1$ can be removed when $z_2$ is an integer.

38: 9.4 Maclaurin Series

… ►For $z\in ℂ$ ►

9.4.1 $$\mathrm{Ai}\left(z\right)=\mathrm{Ai}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Ai}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Source:

Olver (1997b, p. 54)

A&S Ref:

10.4.2 (with 10.4.4 and 10.4.5)

Permalink:

http://dlmf.nist.gov/9.4.E1

Encodings:

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See also:

Annotations for §9.4 and Ch.9

►

9.4.2 $${\mathrm{Ai}}^{\prime }\left(z\right)={\mathrm{Ai}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Ai}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

Permalink:

http://dlmf.nist.gov/9.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

►

9.4.3 $$\mathrm{Bi}\left(z\right)=\mathrm{Bi}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Bi}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

A&S Ref:

10.4.3 (with 10.4.4 and 10.4.5)

Referenced by:

(9.12.15)

Permalink:

http://dlmf.nist.gov/9.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

►

9.4.4 $${\mathrm{Bi}}^{\prime }\left(z\right)={\mathrm{Bi}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Bi}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right).$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

Permalink:

http://dlmf.nist.gov/9.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

39: 4.7 Derivatives and Differential Equations

… ►

4.7.1 $$\frac{d}{dz}\ln z=\frac{1}{z},$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.1.46

Permalink:

http://dlmf.nist.gov/4.7.E1

Encodings:

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See also:

Annotations for §4.7(i), §4.7 and Ch.4

… ►For a nonvanishing analytic function $f(z)$, the general solution of the differential equation ►

4.7.5 $$\frac{dw}{dz}=\frac{f^{\prime }(z)}{f(z)}$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $f(z)$: non-vanishing analytic function

Permalink:

http://dlmf.nist.gov/4.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(i), §4.7 and Ch.4

… ►

4.7.6 $$w(z)=\mathrm{Ln}\left(f(z)\right)+\text{ constant}.$$

ⓘ

Defines:

$w$: solution (locally)

Symbols:

$\mathrm{Ln}z$: general logarithm function, $z$: complex variable and $f(z)$: non-vanishing analytic function

Permalink:

http://dlmf.nist.gov/4.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(i), §4.7 and Ch.4

… ►When $a^z$ is a general power, $\ln a$ is replaced by the branch of $\mathrm{Ln}a$ used in constructing $a^z$. …

40: 4.19 Maclaurin Series and Laurent Series

… ►

4.19.1 $$\sin z=z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+\mathrm{\cdots },$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.65

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.19.E1

Encodings:

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See also:

Annotations for §4.19 and Ch.4

►

4.19.2 $$\cos z=1-\frac{z^2}{2!}+\frac{z^4}{4!}-\frac{z^6}{6!}+\mathrm{\cdots }.$$

ⓘ

Symbols:

$\cos z$: cosine function, $!$: factorial (as in $n!$) and $z$: complex variable

A&S Ref:

4.3.66

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

… ►

4.19.3 $$\tan z=z+\frac{z^3}{3}+\frac{2}{15}{z}^5+\frac{17}{315}{z}^7+\mathrm{\cdots }+\frac{{(-1)}^{n-1}{2}^{2n}(2^{2n}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ ,

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\tan z$: tangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.3.67

Referenced by:

§4.19

Permalink:

http://dlmf.nist.gov/4.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

►

4.19.4 $$\csc z=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}{z}^3+\frac{31}{15120}{z}^5+\mathrm{\cdots }+\frac{{(-1)}^{n-1}2(2^{2n-1}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ ,

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $!$: factorial (as in $n!$), $n$: integer and $z$: complex variable

A&S Ref:

4.3.68

Referenced by:

§4.33

Permalink:

http://dlmf.nist.gov/4.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

… ►

4.19.6 $$\cot z=\frac{1}{z}-\frac{z}{3}-\frac{z^3}{45}-\frac{2}{945}{z}^5-\mathrm{\cdots }-\frac{{(-1)}^{n-1}{2}^{2n}{B}_{2n}}{(2n)!}{z}^{2n-1}-\mathrm{\cdots },$$ ,

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $!$: factorial (as in $n!$), $n$: integer and $z$: complex variable

A&S Ref:

4.3.70

Permalink:

http://dlmf.nist.gov/4.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.19 and Ch.4

…