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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