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21: 10.10 Continued Fractions

§10.10 Continued Fractions

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22: 10.33 Continued Fractions

§10.33 Continued Fractions

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23: 8.17 Incomplete Beta Functions

… ►However, in the case of §8.17 it is straightforward to continue most results analytically to other real values of $a$, $b$, and $x$, and also to complex values. … ►

§8.17(v) Continued Fraction

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8.17.24 $$I_x(m,n)={(1-x)}^n\sum _{j=m}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{n+j-1}{j}\right)x^j,$$ $m,n$ positive integers; .

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

26.5.26 (The upper limit of summation has been corrected.)

Referenced by:

§8.17(i), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/8.17.E24

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation is the same as Equation (26.5.26) of Abramowitz and Stegun (1964), except that the upper limit of the summation has been corrected to $\mathrm{\infty }$.

Suggested 2011-03-23 by Stephen Bourn

See also:

Annotations for §8.17(vii), §8.17 and Ch.8

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24: 4.25 Continued Fractions

§4.25 Continued Fractions

… ►See Lorentzen and Waadeland (1992, pp. 560–571) for other continued fractions involving inverse trigonometric functions. …

25: 20 Theta Functions

Chapter 20 Theta Functions

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26: 7.9 Continued Fractions

§7.9 Continued Fractions

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27: 8.9 Continued Fractions

§8.9 Continued Fractions

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28: 4.9 Continued Fractions

§4.9 Continued Fractions

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§4.9(i) Logarithms

… ►For other continued fractions involving logarithms see Lorentzen and Waadeland (1992, pp. 566–568). … ►

§4.9(ii) Exponentials

… ►For other continued fractions involving the exponential function see Lorentzen and Waadeland (1992, pp. 563–564). …

29: 1.9 Calculus of a Complex Variable

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§1.9(ii) Continuity, Point Sets, and Differentiation

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Continuity

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Differentiation

… ►Differentiability automatically implies continuity. … ►An arc $C$ is given by $z(t)=x(t)+\mathrm{i}y(t)$, $a\le t\le b$, where $x$ and $y$ are continuously differentiable. …

30: 1.8 Fourier Series

… ►(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda$. … ►If a function $f(x)\in C^2[0,2\pi ]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime }(x)$. … ►Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $\left|f^{\prime \prime }(x)\right|$ are integrable over $(-\mathrm{\infty },\mathrm{\infty })$. …