Tagged Questions
-2
votes
0answers
30 views
Solve $ z= \frac{2a^2}{-4+z} + \frac{2a^4}{(-4+z)^2(-16+z)} + \cdots $.
I am trying to solve $z\in \mathbb{C}$ in terms of $a\in \mathbb{C}$, where
$$
z= \frac{2a^2}{-4+z} + \frac{2a^4}{(-4+z)^2(-16+z)} + \cdots.
$$
I plugged $z= \sum_{k=0}^\infty c_k a^k $ into the ...
1
vote
1answer
144 views
An issue with approximations of a recurrence sequence
By trying to give an approximation to a given recurrence sequence I encountered a problem.
To be more precise I have a method but it fails if the right condition is not met and I wonder how I should ...
2
votes
1answer
33 views
Two-point Taylor expansion with one assymptotic point?
According to this paper, a two-point Taylor expansion can be definied like this:
$$\text{Let }f\left(z\right)\text{ be an analytic function and }z_1 \text{and }z_2\in \mathbb{C}, z_1\neq ...
2
votes
0answers
70 views
Series expansion of a series
I would like to perform an asymptotic expansion of the function $$f(x) = \sum_{n=1}^{\infty}\frac{1}{(nx)^2}K_2(n x),$$ where $K_2(x)$ is the modified Bessel function of the second kind, around $x=0$. ...
0
votes
0answers
42 views
Asymptotically expand Laplace transform
Assume $\max_{a\leq t\leq b}{\phi (t)} =\phi (a)$, $\phi '(a)\neq 0,f(a)\neq 0$ and $f(t)$ has a Taylor expansion about $t=a$.
Use integration by parts to show that $I(x)=\int_a^b{f(t)e^{x\phi ...
3
votes
1answer
35 views
Exercise about MacLaurin's polynomial and small-o
In class the professor wrote the following limit:
$\lim_{x\to 0} \frac{\sinh^2 (x) -x^2}{x^4}$
So he "expanded" (sorry for my English) the MacLaurin's formula for $\sinh x$ up to the 3rd power, and ...
0
votes
2answers
60 views
Asymptotic Expansion
I was trying to solve and ODE and while doing some asymptotics, I bumped into something like this $\left(1+\frac{\gamma}{z_{0}}+\epsilon \frac{z_{1}}{z_{0}}\right)^{-2}$
where $\gamma$ $\,$ is of ...
