New answers tagged

9

Tan[(3 π)/11] + 4 Sin[(2 π)/11] // ToRadicals // FullSimplify Sqrt[11]


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 ...


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]


2

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


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 }. ...


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]...


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

$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

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 ...


0

Try the following trick: realPartRule = Complex[re_, im_] :> Complex[re, 0]; realPart[exp__] := exp /. realPartRule; Applying this trick to your result we obtain: realPart[Integrate[Exp[a*(x^3)], {x, 0, b}, Assumptions -> {a > 0, b > 0}]] (*(Gamma[1/3] - Gamma[1/3, -a b^3])/(6 a^(1/3))*)


0

Here is dirty trick suitable for that particular case: rez = Integrate[Exp[5*(x^3)], {x, 0, 7}, Assumptions -> {a > 0, b > 0}] // ToRadicals rezComplex = Integrate[Exp[a*(x^3)], {x, 0, b}, Assumptions -> {a > 0, b > 0}] realAnswer = (rez // ToRadicals) /. {5 -> a, -1715 -> -a*b^3} FullSimplify[(realAnswer - rezComplex), ...


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}} *)


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}}*)


0

Not sure this is what you want, but here is a way to bring it into a simple form: a = (24 (15 + 4 Sqrt[3] x - 4 x^2 + x^4))/((3 + Sqrt[3] x)^4 (29 - 2 x^2 + x^4)) + (64 (58 + 165 x^2 + 28 x^4 + x^6))/(-87 + 35 x^2 - 5 x^4 + x^6)^2 b = Denominator[a[[2]]] c = Numerator[a[[2]]] d = a[[1]] + c/Factor[b] e = Together[d] f = Numerator[e] g = ...


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[0``10. n] Sinh[2 0``10. n] (* 1 *) {Csch[0``10. n], Sinh[2 0``10. n]} // FullForm (* List[Csch[Times[0``10.,n]],Sinh[Times[0``9.698970004336019,n]]] *) (The coefficients are ...


0

Try this: expr = (58500000 (281 E^(-750000000000 t/281) \[Pi] - 281 \[Pi] Cos[10 \[Pi] t] + 75000000000 Sin[10 \[Pi] t]))/(5625000000000000000000 + 78961 \[Pi]^2); Then expr1 = expr /. Exp[Rational[a_, b_]*t] -> 0 /. a_*Cos[10 \[Pi] t] + b_*Sin[10*\[Pi]*t] -> Sqrt[a^2 + b^2] Sin[10*\[Pi]*t + ArcTan[a, b]] (* -((58500000 Sin[...


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^...


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