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Results tagged with calculus-and-analysis
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user 53677
Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.
3
votes
How to quickly calculate the limit of integral with parameters
L'Hôpital's rule needs ~10% of the calculation time
Normal[Series[Integrate[t*Log[1 + t*Sin[t]], {t, 0, x}] , {x, 0, 4}]]/Normal[Series[(1 - Cos[x^2]) , {x, 0, 4}]]
(* 1/2*)
- 62.9k
2
votes
Accepted
How to use L'Hospital's rule for abstract functions
L'Hospital's rule is applicable if the Limit would give a singular expression 0/0
Try
Limit[(a*f[h] + b*f[2 h] - (a + b) f[0])/h, h -> 0, Analytic -> True]
(*(a + 2 b) Derivative[1][f][0]*)
- 62.9k
1
vote
Solving Integral in analytic form takes too long
Use some Assumptions to get a fast evaluation. For simplified integrand (x^2+y^2->x^2, y-h->y) Mathematica evaluates
FullSimplify[Integrate[1/Sqrt[x^2 + (y - t)^2 ], {t, -c, c},
Assumptions -> {c > 0 …
- 62.9k
2
votes
Accepted
How to verify the integral remainder of this integral formula?
Answer to your first question:
Series[Integrate[f[x], {x, 0,h}] - (h/2 (f[0] + f[h]) + \[Alpha] h^2 (f'[0] - f'[h])), {h, 0,3}]
(* f''[0](\[Alpha]-1/12)h^3+O[h^4]*)
Optimal \[Alpha]==1/12
The answer …
- 62.9k
1
vote
How to calculate the derivative of the solution of DSolve?
For a simple solution try DSolveValue
diffPD = {2/r*cA'[r] + cA''[r] == \[Phi]^2/R^2*cA[r], cA[R] == cAR, cA'[0] == 0}
solPD = DSolveValue[diffPD, cA, r]
the solution of the ode is solPD[r] and the d …
- 62.9k
2
votes
Why doesn't Mathematica integrate this?
Mathematica has problem to handle the Abs inside the argument of DiracDelta (Thanks to the answer of @MichaelE2)
Knowing DiracDelta[v-Abs[x]]=DiracDelta[v-x]+DiracDelta[v+x] Mathematica evaluates the …
- 62.9k
0
votes
Why doesn't Mathematica integrate this?
final answer
DiracDelta[v-Abs[x]]==DiracDelta[Abs[x]-v] only contributes to the integral if the argument v-Abs[x] becomes zero. That's only possible if v>=0!
The integral evaluates to
Integrate[A*Exp[ …
- 62.9k
2
votes
How to interpret these integrations with DiracDelta?
The integral exists and the first result is ok.
Obviously Mathematica can't transform the nonlinear argument y-x^2 in your second approach Integrate[DiracDelta[y - x^2], {y, 0, 2}, {x, -1, 1 - y}]. Th …
- 62.9k
4
votes
Solve command does not solve this equation!
supplement to @Bob Hanlon 's answer
ContourPlot shows the possible solution directly:
ContourPlot[f[n, x] == 0, {n, 0, 15}, {x, -5, 5}, MaxRecursion -> 4, FrameLabel -> Automatic]
addendum
The …
- 62.9k
1
vote
Complicated integration
What about NIntegrate (with parameter f2g=f[z]^2 g[z]) :
int[f2g_?NumericQ] :=NIntegrate[(x*Exp[-2*x]*(y^(-0.5)*Exp[-2*y]))*Exp[-(E1*Exp[-x]*Exp[-y]/(f2g))], {x, 0, Infinity}, {y, 0,Infinity}]
Plot[ …
- 62.9k
2
votes
Challenging integral for Gaussian average
If a numerical solution is ok try
int[a_?NumericQ, b_?NumericQ] := NIntegrate[Exp[-x^2/2]/Sqrt[2 Pi] Sqrt[1 + a^2 Sinh[b x]^2], {x, -Infinity,Infinity}]
Plot3D[int[a, b], {a, 0, 1}, {b, 0, 2}, AxesL …
- 62.9k
6
votes
Derivative for arbitrary number of times
Try
f[t_] := Exp[k t] Sinh[t/n]/(Cosh[t/n] - a)
and for example the third derivative f'''[0] evaluates like this
Derivative[3][f][0]
(*-(3/((1 - a)^2 n^3)) + 1/((1 - a) n^3) + (3 k^2)/((1 - a) …
- 62.9k
0
votes
Integrate ArcTanh or ArcTan
Try
f = Integrate[ 1/(Sqrt[(a + b^2/(4 c^2)) x^2 - (1 -2/(3 c))] + (b/(2 c) x))
, {x,1, a}, Assumptions -> { a > 1, b > 0, c > 1}]
and Mathematica evaluates an unique solution
(*(1/(4 a c))(-2 b …
- 62.9k
6
votes
Accepted
Mathematica doesn't give an answer to what wolframalpha answers
Mathematica Sum just returns input means MMA doesn't find an analytical answer
The sum seems to be convergent
SumConvergence[(1/n)^(n), n]
(* True *)
Numerically result
NSum[(1/n)^(n), {n, 1, Infin …
- 62.9k
2
votes
Newton Method with a Parameter in the equation
Perhaps it's easiier to use NestList?
f[x_] := Exp[-x] - a x
NestList[# - f[#]/f'[#] &, 0, 3] // Simplify
%/.a->1. (*{0, 0.5, 0.566311, 0.567143}*)
- 62.9k