# Tag Info

7

TrigToExp does the job Assuming[a > 0, Asymptotic[TrigToExp[Sech[a *x]], x -> Infinity]] 2 E^(-a x) Assuming[a \[Element] Reals, Asymptotic[TrigToExp[Sech[x/a] + Sech[x*a]], x ->Infinity]] ConditionalExpression[2 E^(-(x/a)), a > 1 && a^2 < 3]

6

There is unlikely to be a "nice" closed-form solution for this equation, since it involves both transcendental and algebraic expressions for $v$. However, it can be solved in terms of special functions with a bit of work. We can make the substitution $v = \alpha x + b$ to rewrite the equation (after some algebra) as  x e^x < - e^{-b/\alpha }. ...

4

An alternative answer using components for the vectors: xi = {x1, x2, x3}; H = {H1, H2, H3}; B = Cross[xi, H]; b = B.B; (H.xi)^2*(xi.xi) - ((H.H) (xi.xi)^2 - b (xi.xi)) // FullSimplify returns 0 hence the identity is true. In principle the vectors could also be complex as the components are unspecified.

3

ClearAll["Global*"] Since there are only three equations for the relations, you can only eliminate three variables. In addition to a and X, e will also remain. repl = Solve[{d/(2 b) == X, (b + c)/(2 b) == X, (e - b)/(2 b) == X}, {b, c, d}][[1]] (* {b -> e/(1 + 2 X), c -> (e (-1 + 2 X))/(1 + 2 X), d -> (2 e X)/(1 + 2 X)} *) mat = {{a ...

3

$Assumptions = xi ∈ Vectors[3, Reals] && H ∈ Vectors[3, Reals] B = Cross[xi, H]; b = B . B; (H . xi)^2*(xi . xi) == (H . H) (xi . xi)^2 - b (xi . xi) // TensorReduce True 3 I think I've traced down the problem. It hinges on two things. An identity: Cosh[x] == Sinh[2 x]/(2 Sinh[x]) // Simplify (* True *) And a questionable auto-simplification: Csch[010. n] Sinh[2 010. n] (* 1 *) {Csch[010. n], Sinh[2 010. n]} // FullForm (* List[Csch[Times[010.,n]],Sinh[Times[09.698970004336019,n]]] *) (The coefficients are ... 2 It appears that Mathematica simply cannot handle the asymptotics of Sech in general. When requesting the first order term, if you give it TrigToExp@Sech[a x]$= \dfrac{2}{e^{-a x}+e^{a x}}\$ it's smart enough to deal with those exponentials. But if you try Asymptotic[TrigToExp@Sech[a x], x -> ∞, Assumptions -> a > 0, SeriesTermGoal -> 2] you'll ...

2

Try this: Simplify[Refine[Asymptotic[Sech[a x], a x -> ∞], Assumptions -> Element[a,Reals]]]

2

It is not an answer. I tried to correct the errors in your code. After that the initial expression is expr1 = Cos[\[Theta]14]^3 Sin[\[Theta]13]^2 Sin[\[Theta]14] Sin[\ \[Theta]24] Exp[-i (\[Delta]14 + \[Delta]24 + 2 \[Delta]13)] - Sin[\[Theta]14]^2 Cos[\[Theta]14] Sin[\[Theta]13] Exp[-i \ (\[Delta]13 + 2 \[Delta]14)] (Cos[\[Theta]24] Sin[\[Theta]...

2

Try Collect Solve[-1 + d - d E^-az + f[z]/f0 == 0, f[z]] //Collect[#, {f0, Exp[z_]}] & (*{{f[z] -> (1 - d + d E^-az) f0}}*)

2

May be Clear["Global*"] expr = (58500000 (281 E^(-750000000000 t/281) π - 281 π Cos[10 π t] + 75000000000 Sin[10 π t]))/(5625000000000000000000 + 78961 π^2); expr = Expand[expr]; expr2 = Simplify[If[MatchQ[#,_.*Exp[__*t]],Limit[#,t->Infinity],#]&/@expr]; (Expand@Numerator[expr2]/.a_.*Cos[w_ t]+b_.*Sin[w_ t] :>Sqrt[a^...

2

I propose an initial manual simplification. Replace 1 + s^2 - v^2 by r^2, which is best done by replacing v^2: v^2 -> 1 + s^2 - r^2 You can use Reduce to get new assumptions —— or you can just bite the bullet and think :) —— I did the former: Reduce[r > 0 && r^2 == 1 + s^2 - v^2 && 0 < v < 1 && -v < s < v, {v}, ...

1

Another way: Solve[-1 + d - d E^-az + f[z]/f0 == 0, f[z]] // Simplify[#, ComplexityFunction -> (LeafCount[#] +Count[#, _Symbol, Infinity]&) ]& (* {{f[z] -> (1 + d*(-1 + E^(-az)))*f0}} *)

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