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1: 5.21 Methods of Computation

… ►For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). …

2: About the Project

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Figure 1: The Editors and 9 of the 10 Associate Editors of the DLMF Project (photo taken at 3rd Editors Meeting, April, 2001). The front row, from left to right: Ronald F. …The back row, from left to right: William P. …

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3: 13.12 Products

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4: 13.5 Continued Fractions

… ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side everywhere in $ℂ$. … ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector . …

5: 13.17 Continued Fractions

… ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side for all $z\in ℂ$. … ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector . …

6: Philip J. Davis

… ►In 1957, Davis took over as Chief, Numerical Analysis Section when John Todd and his wife Olga Taussky-Todd, feeling a strong pull toward teaching and research, left to pursue full-time positions at the California Institute of Technology. … ►Davis left NBS in 1963 to become a faculty member in the Division of Applied Mathematics at Brown University, but during the early development of the DLMF, which started in 1998, he was invited back to give a talk and speak with DLMF project members about their plans. …

7: 5.10 Continued Fractions

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5.10.1 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)+z-\left(z-\frac{1}{2}\right)\ln z-\frac{1}{2}\ln \left(2\pi \right)=\frac{a_0}{z+}\frac{a_1}{z+}\frac{a_2}{z+}\frac{a_3}{z+}\frac{a_4}{z+}\frac{a_5}{z+}\mathrm{\cdots },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $z$: complex variable and $a_k$: coefficient

Referenced by:

Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.10.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.10 and Ch.5

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8: 10.70 Zeros

… ►In the case $\nu =0$, numerical tabulations (Abramowitz and Stegun (1964, Table 9.12)) indicate that each of (10.70.2) corresponds to the $m$th zero of the function on the left-hand side. …

9: 19.10 Relations to Other Functions

… ►In each case when $y=1$, the quantity multiplying $R_C$ supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. …

10: 2.3 Integrals of a Real Variable

… ►assume $a$ and $b$ are finite, and $q(t)$ is infinitely differentiable on $[a,b]$. … ►provided that the integral on the left-hand side of (2.3.9) converges for all sufficiently large values of $x$. … ►Without loss of generality, we assume that this minimum is at the left endpoint $a$. … ►When the parameter $x$ is large the contributions from the real and imaginary parts of the integrand in … ►For the more general integral (2.3.19) we assume, without loss of generality, that the stationary point (if any) is at the left endpoint. …