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3

The problem can be reduced (I think) to the problem of defining a phase function $\arg(f(x))$ that is continuous on the interval $x \in [0, 1]$. This is tricky, because the path $f(x)$ may "wind" around the origin, leading to different values of the "continuous argument" for the same value of the "conventional argument". A ...

2

ComplexExpand is what you want. It tells Mathematica to treat all symbols as real unless specified otherwise. In order to allow f[x] to be complex for arbitrary arguments you could define myComplexExpand[expr_] := Module[{g}, ComplexExpand[ expr /. f[x_] /; x \[Element] Reals :> g[x], f[_] ] /. g[x_] :> f[x] ] This then gives reasonable ...

1

Following code does the job. ClearAll[realFunctions, assumptions, re, im]; realFunctions = {f}; assumptions = Element[x, Reals]; re[expr_] := With[{ functions = Reap[Scan[If[MemberQ[realFunctions, Head[#]], Sow[#]] &, expr, {0, \[Infinity]}]][[2, 1]] /. f[a_] /; UnsameQ[True, Refine[Element[a, Reals]/. Thread[realFunctions -> ...

1

Maybe adding Simplify gives you what you want. ComplexExpand[Sqrt[1 - Abs[x]^2]] // Simplify (*Piecewise[{{I*((x^2 - 1)^2)^(1/4), Abs[x] > 1}}, ((x^2 - 1)^2)^(1/4)]*) or $Assumptions = -1 < x < 1 ComplexExpand[Sqrt[1 - Abs[x]^2]] // Simplify (*Sqrt[1 - x^2]*) The plus case is different because the value inside the Sqrt is always positive. It is ... 1 Another possibility using FeynCalc 9.3 or above (could be interesting for particle physicists) CLC[i, j, k] CSI[j].CSI[k] // PauliSimplify[#, PauliReduce -> True] & // FCE 2 I CSI[i] CSI[i].CSI[i] // PauliSimplify // FCE 3 CSI is a shortcut for a Cartesian Pauli sigma matrix, while CLC denotes a Cartesian Levi-Civita tensor. PauliSimplify handles ... 1 Since$\Re(z^2)=x^2-y^2$and$\Im(z^2)=2xy$, the mapping$f(z)=z^2$is equivalent to the mapping$u=x^2-y^2,v=2xy$, so we can plot it by using the ParametricPlot xy = ParametricPlot[{x, y}, {x, -4, 4}, {y, -4, 4}, MeshFunctions -> {#1^2 - #2^2 &, 2 #1*#2 &}, Mesh -> 8, MeshShading -> {{LightRed, LightGreen}, {LightBlue, LightYellow}},... 1 There are ComplexPlot and ComplexPlot3D commands since version 12 of Mathematica. See http://reference.wolfram.com/language/ref/ComplexPlot.html and http://reference.wolfram.com/language/ref/ComplexPlot3D.html for more info. 1 First, let's look at what happens if we factor the ep term out of the integral, like this ClearAll[a, b, f, R, t] ep = a + I b; B = Integrate[f[t] , {t, -R, R}] ep; With[{$Assumptions = Element[Integrate[ f[t], {t, -R, R}], Reals]}, Re[B] // ComplexExpand // Simplify ] (* a*Integrate[f[t], {t, -R, R}] *) So, it looks like MMA is able to apply the ...

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