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Results tagged with ca.classical-analysis-and-odes
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user 25619
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
4
votes
Why is the Laplacian ubiquitous?
The Laplacian is a second order differential operator which is both linear
and invariant under any rigid body motion (rotations and translations).
There is no simpler non-trivial operator with these …
- 664
1
vote
Nonvanishing of Jacobians implies global injectivity?
The answer is NO. Consider the two-dimensional example $F=(f,g)$ with
$$
f(x,y) = \sqrt{2}e^{x/2}\cos(ye^{-x})
$$
$$
g(x,y) = \sqrt{2}e^{x/2}\sin(ye^{-x})
$$
The determinant of the Jacobian matrix is …
- 664
0
votes
Asymptotic estimates for the Exponential
This is a standard calculus question. Let $z(n) = (1−f(n))^{g(n)}$.
Then $\log z(n) = g(n) \log( 1 - f(n))$. Assuming $f(n)\rightarrow 0$, one
has $\log z(n)/( f(n)g(n) ) \rightarrow -1$. For your
p …
- 664