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1: 3.1 Arithmetics and Error Measures

… ►A nonzero normalized binary floating-point machine number $x$ is represented as …where $s$ is equal to $1$ or $0$, each $b_j$, $j\ge 1$, is either $0$ or $1$, $b_1$ is the most significant bit, $p$ ($\in \mathbb{N}$) is the number of significant bits $b_j$, $b_{p-1}$ is the least significant bit, $E$ is an integer called the exponent, $b_0.b_1{b}_2\mathrm{\dots }{b}_{p-1}$ is the significand, and $f=\mathrm{.}{b}_1{b}_2\mathrm{\dots }{b}_{p-1}$ is the fractional part. …

2: 5.10 Continued Fractions

§5.10 Continued Fractions

►For $\mathrm{\Re }z>0$, …

3: 1.12 Continued Fractions

… ►

§1.12(iv) Contraction and Extension

… ►The odd part of $C$ exists iff $b_{2k+1}\ne 0$, $k=0,1,2,\mathrm{\dots }$, and up to equivalence is given by … ►and the even and odd parts of the continued fraction converge to finite values. …

4: 1.15 Summability Methods

… ►For $\mathrm{\Re }\alpha >0$ and $x\ge 0$, the Riemann-Liouville fractional integral of order $\alpha$ is defined by … ►

1.15.48 $${𝐼}^{\alpha }{𝐼}^{\beta }={𝐼}^{\alpha +\beta },$$ $\mathrm{\Re }\alpha >0$, $\mathrm{\Re }\beta >0$.

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Symbols:

$\mathrm{\Re }$: real part and ${𝐼}^{\alpha }$: fractional integral

Referenced by:

§1.15(vi)

Permalink:

http://dlmf.nist.gov/1.15.E48

Encodings:

pMML, png, TeX

See also:

Annotations for §1.15(vi), §1.15 and Ch.1

… ►For , $n$ an integer, and $x\ge 0$, the fractional derivative of order $\alpha$ is defined by …

5: 7.18 Repeated Integrals of the Complementary Error Function

… ►

§7.18(v) Continued Fraction

►

7.18.13 $$\frac{{\mathrm{i}}^n\mathrm{erfc}\left(z\right)}{{\mathrm{i}}^{n-1}\mathrm{erfc}\left(z\right)}=\frac{1/2}{z+}\frac{(n+1)/2}{z+}\frac{(n+2)/2}{z+}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Re }$: real part, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/7.18.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(v), §7.18 and Ch.7

…

6: 7.9 Continued Fractions

§7.9 Continued Fractions

►

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 (in different form)

Referenced by:

§7.9

Permalink:

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

Encodings:

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See also:

Annotations for §7.9 and Ch.7

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7.9.2 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{2z}{2z^2+1-}\frac{1\cdot 2}{2z^2+5-}\frac{3\cdot 4}{2z^2+9-}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

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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

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

►

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 (in different form)

Permalink:

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

Encodings:

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See also:

Annotations for §7.9 and Ch.7

…

7: 5.9 Integral Representations

… ► $\mathrm{\Re }\nu >0$, $\mu >0$, and $\mathrm{\Re }z>0$. (The fractional powers have their principal values.) … ►

5.9.2_5 $$\frac{1}{\mathrm{\Gamma }\left(z\right)}=\frac{{\mathrm{e}}^z{z}^{1-z}}{2\pi }{\int }_{-\pi }^{\pi }{\mathrm{e}}^{-z\mathrm{\Phi }(t)}dt,$$ $\mathrm{\Re }z>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable and $\mathrm{\Phi }(t)$: function

Source:

Temme (1996b, (3.38), p. 71)

Referenced by:

§5.9(i), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/5.9.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

… ►

5.9.3 $$c^{-z}\mathrm{\Gamma }\left(z\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|t|}^{2z-1}{\mathrm{e}}^{-c{t}^2}dt,$$ $c>0$, $\mathrm{\Re }z>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§5.9(i)

Permalink:

http://dlmf.nist.gov/5.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

… ►For $\mathrm{\Re }z>0$, …

8: 5.12 Beta Function

… ►In this section all fractional powers have their principal values, except where noted otherwise. In (5.12.1)–(5.12.4) it is assumed $\mathrm{\Re }a>0$ and $\mathrm{\Re }b>0$. … ►In (5.12.8) the fractional powers have their principal values when $w>0$ and $z>0$, and are continued via continuity. … ►In (5.12.11) and (5.12.12) the fractional powers are continuous on the integration paths and take their principal values at the beginning. …when $\mathrm{\Re }b>0$, $a$ is not an integer and the contour cuts the real axis between $-1$ and the origin. …

9: Wadim Zudilin

… ►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. …

10: 10.22 Integrals

… ►

Fractional Integral

… ►When $\mathrm{\Re }\mu >-1$ … ►When $\mathrm{\Re }\mu >0$, … ►When $\mathrm{\Re }\nu >\mathrm{\Re }\mu >-1$, … ►When $\mathrm{\Re }\mu >0$, …