Questions on dealing with series data and constructing power series expansions of functions.

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0
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1answer
455 views

Asymptotic expansion, negative powers

The question was inspired by this discussion: How to expand a function into a power series with negative powers? I am interested in asymptotic behavior of a function at infinity: ...
8
votes
0answers
191 views

Why does $\frac{\partial}{\partial x}O\left(\left(\frac{1}{x}\right)^0\right)$ equal $O\left(\left(\frac{1}{x}\right)^0\right)$ in a series expansion?

When taking the derivative of a series expansion around a finite point, the $O(x^n)$ part is differentiated as expected. $O(x^n)$ becomes $O(x^{n-1})$ except $O(x^0)$ which stays $O(x^0)$. When ...
7
votes
2answers
1k views

How to study asymptotic behavior, built-in functions

My question is as follows. Suppose we have a function $f(r)$ and we want to study its asymptotic behavior at infinity ($r\rightarrow \infty$). For example, the function may reduce to $-\frac{a}{r}$ or ...
6
votes
1answer
699 views

How to expand a function into a power series with negative powers?

Is there any way to expand this expression a+b(1-Exp[-T/(b c)]/(z-Exp[-T/ (b c)]) (where a, ...
9
votes
3answers
2k views

Multivariable Taylor expansion does not work as expected

The basic multivariable Taylor expansion formula around a point is as follows: $$ f(\mathbf r + \mathbf a) = f(\mathbf r) + (\mathbf a \cdot \nabla )f(\mathbf r) + \frac{1}{2!}(\mathbf a \cdot ...
1
vote
1answer
501 views

How can I get a Taylor expansion of the Sin[x] function?

How can I get a Taylor expansion of the Sin[x] function by the power series?
2
votes
2answers
662 views

About generating power series

For an arbitrary function $f(x,y)$ I am defining functions LogMT1 and LogMT2 as follows, ...
2
votes
1answer
258 views

About high dimensional integrals

I want to be able to do high dimensional integrals like, (..naively I wrote it as this..) ...
8
votes
2answers
342 views

Series expansion with irrational power

I need the series expansion of a fairly nasty function and its derivative: ...
3
votes
3answers
417 views

Limiting form of a polynomial expression

When simplifying an expression by hand, a trick that is often used is to remove terms that are lower powers of the independent variable, for instance, as $x \rightarrow \infty$, $x^2 + x$ becomes ...
0
votes
1answer
619 views

Fourier series of interpolating function result of NDSolve

I am having a tough time formulating the right question but here goes. I know that solving the pde as in here gives me an interpolating function. I understand that the interpolating function object ...
6
votes
1answer
173 views

Sophistication of Series[…]

I'll give a concrete example and I hope that my general question will be clear. Say I have three variables, $f$, $g$, $h$, and I know that $f=\mathcal O(x)$, $g=\mathcal O(x^2)$, $h=\mathcal O(x^3)$ ...
9
votes
2answers
560 views

Series expansion of an inverse

I have to find the series expansion of the inverse function of : $\arctan\left(\frac{\ln(1+x)}{1+x}\right)$ How do I find out the series expansion of any inverse ? Note: The inverse of a function ...