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Solutions to algebraic or transcendental equations are expressed in terms of Root objects whenever it is impossible to find explicit solutions. In general there is no way express roots of 5-th (or higher) order polynomials in terms of radicals. However even higher order algebraic equations can be solved explicitly if an associated Galois group is solvable. ...

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I guess that V9 now adds this capability: $Assumptions = { Element[A, Matrices[{m, n}]], Element[B, Matrices[{n, k}]] }; TensorReduce[ Transpose[Transpose[A].Transpose[B]] ] (* Out: B.A *) 20 The most direct way to test this is probably the following:$Assumptions = x > 0; Element[x, Reals] // Simplify (* Out[1]= True *) $Assumptions = True; Element[x, Reals] // Simplify (* Out[4]= x ∈ Reals *) So$x>0$seems to imply that$x$is real. 20 You can't use replacements that way, because Mathematica does not do replacements on expressions the way they appear to you. To see what I mean, take a look at the FullForm of your expression: x/(y*z) // FullForm Out[1]= Times[x,Power[y,-1],Power[z,-1]] Whereas, the replacement that you're using is Times[y, z]. In general, it is not a good idea to use ... 18 Mathematica does not support this directly. You can do things of this sort using an external package called NCAlgebra. http://math.ucsd.edu/~ncalg/ The relevant documentation may be found at http://math.ucsd.edu/~ncalg/DOWNLOAD2010/DOCUMENTATION/html/NCBIGDOCch4.html#x8-510004.4 In particular have a look at "4.4.8 NCLDUDecomposition[aMatrix, Options]" ... 18 What you have is a MultinormalDistribution. The quadratic and linear forms in the exponential can be rewritten in terms of$\frac12(\vec{x}-\vec{\mu})^\top\Sigma^{-1}(\vec{x}-\vec{\mu})$where$\vec{\mu}$represents the mean and$\Sigma$the covariance matrix, see the documentation. With this, you can do integrals of the type given in the question by ... 17 It is assumed that$x$is a real number. Everything else would mathematically not make sense because on complex numbers there does not exist an ordering relation. An example would be to take the expression$\sqrt{x^2}$and to imagine that this is not equal$x$for$x=-\mathbb{i}$. Therefore the expression is in a general form not simplified In[37]:= ... 17 The nearest Mathematica has to "types" are Heads of expressions that are Atoms. For example: Through[{AtomQ, Head}[2]] {True, Integer} Through[{AtomQ, Head}[2 + I]] {True, Complex} Through[{AtomQ, Head}["cat"]] {True, String} and so on... There are also somewhat different "types" in the context of Compile. 17 In this case you can use SeriesCoefficient SeriesCoefficient[Exp[x], {x, 0, n}] 15 Because the assumption system is not called during the standard evaluation sequence, it is only called when Simplify, FullSimplify, Sum, Integrate etc... are used. Thus, x>0 remains unevaluated: Assuming[x > 0, x > 0] (* ==> x > 0 *) and TrueQ then returns False: Assuming[x > 0, TrueQ[x > 0]] (* ==> False *) If, however, you ... 15 There is no need to play around with ReplaceAll, Rule, Block, Module or whatever using D, since you have an oparator Derivative really fulfilling your needs while you need not bother if the arguments were defined, so I recommend it to find symbolic derivatives of your function. Remember of shorthands f', f'' to represent first and second derivatives of ... 15 Regarding your general question: from my own experience, a large part of my research findings while still at academia would have been much harder or outright impossible to get without Mathematica, and that applies to both numerical and analytical work. I also know this for many other people. There are a number of areas where it allows you to get through some ... 15 If you use the third argument in Solve, i.e. a list of variables to be eliminated (take a look at the Eliminating Variables tutorial in Mathematica) then you'll get the result immediately : Solve[{a b c == -1, a^2/c + b/c^2 == 1, a^2 b + b^2 c + c^2 a == t, a b^5 + b c^5 + c a^5 == res}, {res}, {a, b, c, t}] {{res -> 3}} Edit ... 15 Looking at the Trace of one which does work: x = Sin[Pi/5] (* Sqrt[5/8 - Sqrt[5]/8] *) Trace[ArcSin[x], TraceInternal -> True] It appears that Mathematica computes the ArcSin numerically and then recognises the result, 0.628319 as possibly equal to Pi/5. To check it computes Sin[Pi/5], and subtracts it from the original argument to see if it gets ... 14 You are assuming that $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$ This is not generally true. Take for example$a=1$and$b=-1$for which this identity does not hold. You need to give additional assumptions to Simplify, in this case that$b>0$. Simplify[Sqrt[a/b] == Sqrt[a]/Sqrt[b], b > 0] (* ==> True *) 14 Removing the imaginary portion of an expression is done by doing ComplexExpand[Re[expression]]. Using just Re alone will not work as Re does no evaluation on symbols with unknown complex parts. Now as stated in the problem and the comments above this particular problem requires a fair amount of assumptions. The simplest way to add local assumptions is to ... 14 In general, one cannot get explicit analytic solutions of trancendental equations in terms of radicals. This is also the case of univariate higher order polynomial equations. On the other hand since Mathematica 7 we can find exact solutions (in terms of Root objects) of a wide range of (univariate) trancendental equations, for more detailed discussion of ... 14 Short story $$\vartheta(x) = \arg \left[(\operatorname{Bi}x+i \operatorname{Ai}x)e^{-\frac{2}{3} i (-x)^{3/2}}\right]+\frac{2}{3} \operatorname{Re}\left[(-x)^{3/2}\right]$$ Update: I see that you want use only real functions, so you can expand this as $$\vartheta(x) = \begin{cases} \arctan\frac{\cos \left(\frac{2}{3} (-x)^{3/2}\right) ... 14 A first step would be to implement a convenience function that can automatically apply the method of separation of variables to separable types of equations. To show that the steps could in principle be automated, let me repeat basically the same calculation that I did for cylindrical coordinates with only slight modifications to the heat equation: ... 13 An experimental internal function IntegrateInverseIntegrate helps here, although it's intended more for integrands involving logs. This is what it returns in the development version: IntegrateInverseIntegrate[Exp[-x Cosh[t]], {t, 0, Infinity}, Assumptions -> Re[x] > 0] (* BesselK[0, x] *) 13 Use the following representation of the Legendre polynomials:$$ P_n(x) = 2^n \sum_{k=0}^n x^k \binom{n}{k} \binom{\frac{n+k-1}{n}}{n} $$Note that the sum effectively is over k \equiv n \bmod 2. Expand each Legendre polynomial into a sum. Integration with respect to \theta is easy:$$ \int_0^{\pi} \sin^{k_1+k_2+k_3+1} \theta \mathrm{d}\theta ... 13 Since nobody pointed this out I think there is still room for another reply. Note that this works fine Unevaluated[(x + Log[y*z])/(y*z)] /. (y*z) :> w (x + Log[w])/w In more complex cases you may also need to use HoldPattern Unevaluated[(x + Log[(y*z)/2])/((y*z)/2)] /. HoldPattern[((y*z)/2)] :> w (x + Log[w])/w This is not a panacea. ... 13 The code for the default ComplexityFunction was posted on MathSource a number of years ago by Adam Strzebonski (of Wolfram Research). You will see reference to the original reply from Adam referenced in a MathGroup reply from Andrzej Kozlowski dated 12 Jan 2010 with the subject: "[mg106386] Re : Radicals simplify". I mention all that because I can't get the ... 13 This is not an answer (yet). Rather it explores the question in more depth. n = 8; parameters = ConstantArray[{0, 1}, n]; variables = Symbol /@ CharacterRange["a", FromCharacterCode[ToCharacterCode["a"] + n - 1]]; The following takes a long time to evaluate, but the results it produces reveal give us a better view of the problem with Probability. ... 13 Here's the exact answer: i1 = Integrate[x^n Exp[-(x - a)^2], {x, 0, Infinity}, Assumptions -> n > 0] /. n -> 1/2 (* 1/2 E^-a^2 (Gamma[3/4] Hypergeometric1F1[3/4, 1/2, a^2] + 1/2 a Gamma[1/4] Hypergeometric1F1[5/4, 3/2, a^2]) *) i1 /. a -> 0.3 (* 0.907605 *) 13 Working with such a sophisticated function as Reduce, if we can't get the result initially we should add possibly many assumptions. Without the Backsubstitution option it yielded: Reduce[ Abs[x] + Abs[y] + Abs[z] + Abs[t] == 1 && t != 0, {x, y, z, t}, Reals] No more memory available. Mathematica kernel has shut down. Try quitting other ... 12 There is nothing wrong with the issue in the question. Mathematica shouldn't evaluate Simplify[ Integrate[f[x] + g[x], x] == Integrate[f[x], x] + Integrate[g[x], x]] to True, because in general such a rule would be mathematically simply wrong. Consider e.g.$ \forall_{x } f(x) = - g(x)$, while$f$is not e.g. Lesbegue integrable. Of course ... 12 While the answer of Szabolcs is clearly the best alternative, if you have already assigned values to the variables and clearing them for some reason is no viable option, you can use Block[{s, L0, L1, a}, Hold@Evaluate@D[L[s, L0, L1, a], s]] Unlike Module, Block doesn't introduce new variable names but temporarily removes the values of those given. Hold ... 12 Firstly convert the equations to matrix/vector form (i.e.$m_1.x=-m_0\$). {m0, m1} = CoefficientArrays[Equations, Unknows] Then compute the (symbolic) solution. soln = LinearSolve[m1, -m0] // Chop[#, 10^-6] & I have chopped parts of the solution that are small, but you don't have to do this. Verify that the solution is correct. m1.soln + m0 // ...

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Working with RSolve we can find much more than only a few first terms, here is a general term of your function u[n] e.g. : u[n_] = u[n] /. Flatten[ RSolve[{ u[1] == 1, u[2] == 2, u[3] == 3, u[n] == -u[n-3] + 3 u[n-2] + 2 u[n-1]}, u[n], n]] Root[1 - 3 #1 - 2 #1^2 + #1^3 &, 3]^n Root[-45 + 457 #1 - 1028 #1^2 + 257 ...

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