central differences in imaginary direction

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31: 3.10 Continued Fractions

… ►

Quotient-Difference Algorithm

►For several special functions the $S$-fractions are known explicitly, but in any case the coefficients $a_n$ can always be calculated from the power-series coefficients by means of the quotient-difference algorithm; see Table 3.10.1. … ► … ►($\nabla$ is the backward difference operator.) …

32: 7.9 Continued Fractions

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7.9.1 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{z}{z^2+}\frac{\frac{1}{2}}{1+}\frac{1}{z^2+}\frac{\frac{3}{2}}{1+}\frac{2}{z^2+}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

7.1.14 (indifferent form)

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E1

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Annotations for §7.9 and Ch.7

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7.9.3 $$w\left(z\right)=\frac{\mathrm{i}}{\sqrt{\pi }}\frac{1}{z-}\frac{\frac{1}{2}}{z-}\frac{1}{z-}\frac{\frac{3}{2}}{z-}\frac{2}{z-}\mathrm{\cdots },$$ $\mathrm{\Im }z>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

7.1.15 (indifferent form)

Permalink:

http://dlmf.nist.gov/7.9.E3

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See also:

Annotations for §7.9 and Ch.7

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33: Mathematical Introduction

… ►As a consequence, in addition to providing more information about the special functions that were covered in AMS 55, the NIST Handbook includes several special functions that have appeared in the interim in applied mathematics, the physical sciences, and engineering, as well as in other areas. … ►Two other ways in which this Handbook differs from AMS 55, and other handbooks, are as follows. … ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). … ►Similarly in the case of confluent hypergeometric functions (§13.2(i)). … ►In the DLMF this information is provided in pop-up windows at the subsection level. …

34: 7.7 Integral Representations

… ►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. … ►

7.7.2 $$w\left(z\right)=\frac{1}{\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}dt}{t-z}=\frac{2z}{\pi \mathrm{i}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}dt}{t^2-z^2},$$ $\mathrm{\Im }z>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

A&S Ref:

7.1.4 (indifferent form)

Referenced by:

§7.19(i), §7.22(i), §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E2

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

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7.7.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-a{t}^2+2\mathrm{i}zt}dt=\frac{1}{2}\sqrt{\frac{\pi }{a}}{\mathrm{e}}^{-z^2/a}+\frac{\mathrm{i}}{\sqrt{a}}F\left(\frac{z}{\sqrt{a}}\right),$$ $\mathrm{\Re }a>0$.

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Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

7.4.6 7.4.7 (indifferent notation)

Referenced by:

§7.14(i), §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E3

Encodings:

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

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

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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 (indifferent form)

Referenced by:

§7.7(i)

Permalink:

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

Encodings:

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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). …

35: Wadim Zudilin

… ► 1970 in the USSR) is Professor of Mathematics at the University of Newcastle (NSW, Australia). …His research interests are primarily focused on applications of special functions in different parts of number theory. Zudilin is author or coauthor of numerous publications including the book Neverending Fractions, An Introduction to Continued Fractions published by Cambridge University Press in 2014. …He is a member of several editorial boards including the series Monographs in Number Theory published by World Scientific. ►In November 2015, Zudilin was named Associate Editor for the following DLMF Chapter …

36: 28.26 Asymptotic Approximations for Large $q$

… ►

28.26.1 $${\mathrm{Mc}}_m^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fc}}_m(z,h)-\mathrm{i}{\mathrm{Gc}}_m(z,h)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E1

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Annotations for §28.26(i), §28.26 and Ch.28

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28.26.2 $$\mathrm{i}{\mathrm{Ms}}_{m+1}^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fs}}_m(z,h)-\mathrm{i}{\mathrm{Gs}}_m(z,h)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E2

Encodings:

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See also:

Annotations for §28.26(i), §28.26 and Ch.28

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28.26.3 $$\varphi =2h\sinh z-\left(m+\frac{1}{2}\right)\arctan \left(\sinh z\right).$$

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Symbols:

$\sinh z$: hyperbolic sine function, $\arctan z$: arctangent function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.8 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E3

Encodings:

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See also:

Annotations for §28.26(i), §28.26 and Ch.28

►Then as $h\to +\mathrm{\infty }$ with fixed $z$ in $\mathrm{\Re }z>0$ and fixed $s=2m+1$, … ►The asymptotic expansions of ${\mathrm{Fs}}_m(z,h)$ and ${\mathrm{Gs}}_m(z,h)$ in the same circumstances are also given by the right-hand sides of (28.26.4) and (28.26.5), respectively. …

37: 36.15 Methods of Computation

… ►Close to the origin $𝐱=\mathrm{𝟎}$ of parameter space, the series in §36.8 can be used. … ►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. … ►Direct numerical evaluation can be carried out along a contour that runs along the segment of the real $t$-axis containing all real critical points of $\mathrm{\Phi }$ and is deformed outside this range so as to reach infinity along the asymptotic valleys of $\exp \left(\mathrm{i}\mathrm{\Phi }\right)$. …There is considerable freedom in the choice of deformations. … ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\mathrm{\Phi }$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …

38: 1.10 Functions of a Complex Variable

… ►In addition, … ►If $D=ℂ\setminus (-\mathrm{\infty },0]$ and $z=r{\mathrm{e}}^{\mathrm{i}\theta }$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with in both cases. … ►If the path circles a branch point at $z=a$, $k$ times in the positive sense, and returns to $z_0$ without encircling any other branch point, then its value is denoted conventionally as $F((z_0-a){\mathrm{e}}^{2k\pi \mathrm{i}}+a)$. … ►It should be noted that different branches of ${(w-w_0)}^{1/\mu }$ used in forming ${(w-w_0)}^{n/\mu }$ in (1.10.16) give rise to different solutions of (1.10.12). … ►Many properties are a direct consequence of this representation: Taking the $x$-derivative gives us …

39: DLMF Project News

error generating summary

40: 12.14 The Function $W(a,x)$

… ►In this section solutions of equation (12.2.3) are considered. …In other cases the general theory of (12.2.2) is available. … ►The coefficients $c_{2r}$ and $d_{2r}$ are obtainable by equating real and imaginary parts in … ►As noted in §12.14(ix), when $a$ is negative the solutions of (12.2.3), with $z$ replaced by $x$, are oscillatory on the whole real line; also, when $a$ is positive there is a central interval in which the solutions are exponential in character. In the oscillatory intervals we write …