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Results tagged with ca.classical-analysis-and-odes
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user 49208
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
1
vote
Accepted
Derivative in terms of finite differences
This is a special case of well-known general formula for the n-th derivative via differences and Stirling numbers. Cf. good books on finite differences, e.g. of Gelfond or Jordan.
- 1,560
2
votes
Accepted
Estimate of a ratio of two incomplete gamma functions
In fact it seems to be a consequence of the Cauchy theorem from calculus. Really, by it and a formula for derivative of incomplete gamma function (cf. Wiki for example) we evaluate
$\frac{f(x)-f(y)}{ …
- 1,560
15
votes
Sum of series $a^{i^2}$
This sum equals exactly to:
$$ \frac{1}{2}\left(\theta_3 (0,a) -1 \right),$$
and $\theta_3$ is the Jacobi $\theta$ - function.
- 1,560
2
votes
Accepted
A book about almost periodic functions
What do you mean "a new", of what years?
Classics are always new-the books of Bohr, Levitan and Zhikov, Besikovich himself.
There is also a book of Corduneanu with standard name "almost periodic funct …
- 1,560
5
votes
2
answers
607
views
Integrals involving trigonometric functions and polynomials
Can one describe all the real polynomials $P(x)$ such that the following integrals converge:
$$
\int_0^{\infty} \sin(P(x))dx, \int_0^{\infty} \cos(P(x))dx ?
$$
Among special cases are such celebriti …
- 1,560
7
votes
Accepted
A (likely) positivity property of the Lerch zeta-function
From Prudnikov,Brychkov,Marichev Integral and Series, Vol.1, sect. 5.4.3, formula 1 we derive that
$$
\sum_{k=0}^\infty \frac{\cos(ak)}{(k+1/2)^{p+1}}=
\frac{1}{\Gamma(s)}\int_0^{\infty}\frac{t^p e^{- …
- 1,560