12
votes
Series vs Asymptotic in 12.1
Extended comment, I won't accept this as an answer.
Here are some cases I've found where Series might be a better choice than ...
- 988
10
votes
Accepted
How could this asymptotic expansion be obtained?
Exact expression for $\sigma_n$:
b[n_] = BesselJ[1, BesselJZero[0, n]]*BesselJZero[0, n]*StruveH[0, BesselJZero[0, n]];
σ[n_] = π/2*(-1)^n*(b[n + 1] - b[n]);
...
- 43.6k
9
votes
Accepted
7
votes
Accepted
Asymptotic expansion around infinity for inverse cdf of normal distribution
Using assumptions and an immediate assignment:
...
- 43.6k
7
votes
Accepted
AsymptoticSum does not give any output
How about analytically evaluating the product
Product[1 + (3/(4 n)), {n, 2, j}]
$$\frac{\Gamma \left(j+\frac{7}{4}\right)}{\Gamma \left(\frac{11}{4}\right) \Gamma (...
- 17.2k
7
votes
Asymptotic inverse function?
@Roman's answer is exactly what you asked for, but as a check, here is a second answer using Bessel's series expansion for the solution of the Kepler equation:
...
- 11.7k
6
votes
Accepted
Not getting the correct asymptotic behaviour when sending a small parameter to zero
You could try using AsymptoticSolve instead:
...
- 127k
5
votes
Accepted
5
votes
5
votes
Accepted
Determinant of matrix with asymptotic expansion
The O representation of an expansion point of Infinity is obtained with:
O[x, Infinity]
(...
- 127k
5
votes
Accepted
Asymptotic[] Doesn't Actually Compute
tao2 = (2*(-(1/4)/E^(2*t^2) +
Pi/8 - ((1/4)*Sqrt[Pi]*t)/E^t^2 + (1/4)*Pi*Erf[t] -
((1/4)*Sqrt[Pi]*t*Erf[t])/E^t^2 + (1/8)*Pi*Erf[t]^2))/Pi;
...
- 143k
5
votes
Accepted
AsymptoticDSolveValue returing input
The ode
ode = eps y''[x] + (x + x^3)*y'[x] - 2*y[x] == 0
ode /. y->(y[-#]&)
(* -2 y[-x] - (x + x^3) Derivative[1][y][-x] +eps (y^\[Prime]\[Prime])[-x] == 0*)
...
- 44.3k
5
votes
Accepted
Asymptotics and limits of second order ODE which depend on (two) parameters
You could find the solution $h(x)$ and then use Asymptotic on its derivative?
...
- 130k
5
votes
AsymptoticIntegrate of a difficult integral
Edit: there's a factor of 2 missing from the RHS of the $\sin(\sin x)$ relation, which follows from a mistake/typo in the Maths.S.E answer for which I provided a ...
- 12k
5
votes
Accepted
Is there any possibility of obtaining an asymptotic approximation (instead of numerical solutions) of such a 2nd-order homogeneous ODE in Mathematica?
Here's a symbolic solution, whose asymptotic expansion may be obtained:
...
- 228k
5
votes
When to use Series vs Asymptotic?
The definition of Asymptotic can be accessed by GeneralUtilities`PrintDefinitionsLocal, and it seems that part of its code ...
- 1,675
4
votes
Is there a way to check whether $f(x)=o(g(x))$ for given $f$ and $g$?
Another pre V11.3 method is to use the limit definition:
littleO[f_, g_, x_, x0_:Infinity] := PossibleZeroQ[Limit[f/g, x -> x0]]
littleO[x, x^2, x]
...
- 34.7k
4
votes
How to study asymptotic behavior, built-in functions
Another possibility is to use new in M12 function AsymptoticSolve:
sol = AsymptoticSolve[y==f[r],y,{r,Infinity,4}]
...
- 127k
4
votes
Finding Asymptotics for a Series
As @DanielLichtblau says, Mathematica can do the symbolic sum in terms of LerchPhi:
sum = Sum[2^i/i, {i, n}]
-I ([Pi] - I 2^...
- 127k
4
votes
AsymptoticDSolveValue multiple solutions
DSolve does not find $y(x)=x$ either, and I think this is why AsymptoticDSolveValue does not. I am sure they share some core ...
- 130k
4
votes
Accepted
Asymptotic inversion of ExpIntegralEi function
You got correct hints on Math.SE. Define:
E1[z_] := EulerGamma - ExpIntegralEi[-z];
E2[z_] := -ExpIntegralEi[-z] Exp[z];
They can be inverted like this
...
- 17.2k
4
votes
Accepted
Selecting the negative expression
Select negative solutions.
Select[sols,
Resolve[Exists[ϵ, ϵ > 0,
ForAll[s,
Pi - ϵ < s < Pi, (t /. # /. Theta -> s) < 0]]] &]
<...
- 53.9k
4
votes
AsymptoticDSolveValue returing input
Mathematica can not do it. May be because it is boundary value problem. Compare
...
- 130k
3
votes
Accepted
Asymptotic Output Tracking: Code Issues
Updated answer
Use the chain rule and express $f'[t]$ in terms of $x'[t]$. Then the two equations can be solved for $x'[t]$ and $H'[t]$. AffineStateSpaceModel (...
- 8,537
3
votes
Asymptotic behavior of $n^2-n$
Asymptotic[n^2 - n, n -> ∞]
(* n^2 *)
From the documentation:
Asymptotic[expr, x->x0] computes the leading term in ...
- 43.6k
3
votes
Accepted
Does AsymptoticSum work with Arithmetical Number Theoretic Functions?
Although AsymptoticSum appears to not presently have the functionality you require, I expect that it will be added, since the required expansions are known and ...
- 2,742
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