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Results tagged with special-functions
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user 17218
Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.
1
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Bound for truncation error of continued fraction for $E_1(z)$
Let $z \in \mathbb C \setminus(-\infty,0)$. It is known that
$$E_1(z) = \cfrac{e^{-z}}{z+\cfrac{1}{1+\cfrac{1}{z+\cfrac{2}{1+\cfrac{2}{z+\cfrac{3}{1+\cdots}}}}}}.$$
For example, see http://functions. …
- 4,985
7
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Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\ga...
This isn't a full answer but gives another surprising connection between the two constants. One has
$$\gamma = \int_1^\infty \frac{1-\{x\}}{x^2} dx$$
and
$$\log \frac{4}{\pi} = \int_1^\infty \frac{\V …
- 4,985
7
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1
answer
515
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continued fraction for logarithmic integral
Does the logarithmic integral function $\operatorname{li}(x)$ have the continued fraction expansion
$$\operatorname{li}(x) = \cfrac{x}{\log x -1 -{}} \ \cfrac{1}{\log x -3 -{}} \ \cfrac{4}{\log x - …
- 4,985