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Results tagged with special-functions
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user 113397
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.
12
votes
Nicer expression for 2.1369288...?
With a few substitutions, we find that $$c=\frac{k^3}{k^2-k+1}\quad\text{where}\quad1+\frac k{(1-k)^2}=e^k.$$ The solution for $k$ requires a more advanced function than Lambert $W$.
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8
votes
0
answers
291
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Is there a real-analytic approach to evaluate a definite integral (with an elementary integr...
I have never seen a real-analytic approach to evaluate integrals of the form below
$$\int_a^b\text{elementary function}(x)\,dx=\text{constant non-trivially involving}\,W(\cdot)\tag1$$ The elementary f …
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3
votes
1
answer
82
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Curious asymptotics of real part of ratio between Lambert W branches
This question was inspired by the inactive thread How to find this value of $A$? but the focus there was on the divergence of the imaginary part.
It seems that for a given nonzero real $x$, $$\opera …
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4
votes
$\int_L^\infty \exp(- t - y/t) \, dt = \text{?}$
References to the study of these functions which are frequently used in hydrological models. No precise bounds in the following papers but at least they give a starting point.
Harris (2008) "Incomple …
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