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Results tagged with asymptotics
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user 25510
Asymptotic behavior of functions, asymptotic series and related topics
5
votes
Accepted
Asymptotic behavior for the solution of a nonlinear ODE
I believe that a rigorous justification of this asymptotics can be obtained from the papers:
J. Malmquist, Acta Math. 73 (1940), 87–129; 74 (1941), 1–64, 109–128; MR0003898 and
M. …
- 91.8k
7
votes
textbooks on asymptotic expansions
Asymptotics and special functions. Computer Science and Applied Mathematics. Academic Press, New York-London, 1974.
and (an older book)
MR0115035 Ford, Walter B. …
6
votes
Growth of (integral of) Laplace transform of a function of compact support as $Re \to -\infty$
It has exponential growth: if your integral is $O(e^{-\sigma\epsilon}), \sigma\to -\infty,$ then the
support of your function $f$ is contained in $[0,\epsilon]$. This follows
from the inversion formul …
- 91.8k
3
votes
Accepted
Asymptotic behavior of a solution of an ODE
It is possible to solve it explicitly in terms of the Airy function.
Airy's equation in the standard form is
$$y''=xy.$$
Your equation is reduced to this by $x\mapsto-x$ followed by a shift of the ind …
- 91.8k
4
votes
Accepted
Asymptotic Behaviour of Solutions to a Riccati-type ODE with Small Forcing Term
In your example you obtained two linearly independent solutions with different behavior: $\cot(1/r)\sim r,\; r\to\infty$, so your second solution is $O(r^{-2})$.
This is the general pattern if you as …
- 91.8k
2
votes
Reference request: using integral equations to study asymptotics of ODEs
Fedoryuk, Mikhail V. Asymptotic analysis. Linear ordinary differential equations. Springer-Verlag, Berlin, 1993. viii+363 pp. ISBN: 3-540-54810-6
Wasow, Wolfgang Asymptotic expansions for ordinary d …
- 91.8k
9
votes
Accepted
Riemann surface from Riccati equation
The answer to the highlighted question is "no". When $V$ is a polynomial, the
general solution of the Riccati equation is single valued, it is a meromorphic function in the complex plane. To prove the …
- 91.8k
3
votes
Asymptotic behaviour/upper bound for $\int_0^{\infty} \exp(-c x^a+K x^b)dx$ for $a>b>0$ as $...
This is a simple example for the Laplace Method of asymptotic evaluation of integrals.
The essence of the method is that the main contribution to the integral comes from a small
neighborhood of the cr …
- 91.8k
0
votes
Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform
The answer to your question is "no" in a very strong sense: I will construct $a$ such that
$a(x,0)\neq 0$ and $$\int_0^1e^{\lambda x}a(x,1/\lambda)dx\equiv 0.$$
Begin with $a(x)$ infinitely smooth, no …
- 91.8k
2
votes
Accepted
Limit (at infinity) for the lowest eigenvalue of a perturbed harmonic oscillator
Consider two operators $L_1w=-w''+U(x)w$ with eigenvalues $\lambda_k$ and $L_2w=-w''+V(x)w$
with eigenvalues $\mu_k$. If $U\geq V$ then $\lambda_k\geq \mu_k$.
To prove this consider the Rayleigh ratio …
- 91.8k
3
votes
Accepted
Largest asymptotic growth for $2f(x)-f(2x)$
Let us discretise the problem by setting $a_n=2^{-n}f(2^n)$, $b_n=2^{-n-1}\Delta_f(2^n)$. Then your relation becomes,
$$b_n=a_n-a_{n+1}.$$
since $a_n,b_n$ are non-negative, we conclude that
$$\sum_{n= …
- 91.8k
1
vote
Asymptotic behavior of the solution of the high degree differential equation $(x^{2n}y^{(n)}...
Anyway, there is an algorithm of obtaining these asymptotics. …
- 91.8k
2
votes
Proof of Szegö asymptotic theorem
Asymptotics for the zeros of the partial sums of ez. I. Rocky Mountain J. Math. 21 (1991), no. 1, 99–121.
MR1071774 Varga, R. S.; Carpenter, A. J. … Asymptotics for the zeros of the partial sums of ez. II. Computational methods and function theory (Valparaíso, 1989), 201–207, Lecture Notes in Math., 1435, Springer, Berlin, 1990.
T. …
- 91.8k