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I have a real variable $t$ and a function $f(t)$ giving complex values. How to plot the function for $t$ in a given real interval? Also is there a way to do it in Wolfram Alpha website?

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    $\begingroup$ f[t_]:=2 t+I t^2; ParametricPlot[{Re[f[t]],Im[f[t]]},{t,-3,3}] works with Mathematica. ParametricPlot[{Re[2 t+I t^2], Im[2 t+I t^2]},{t,-3,3}] works with WolframAlpha. $\endgroup$ – Bill Dec 13 '20 at 20:26
  • $\begingroup$ You can also use ReIm[f[t]] instead of {Re[f[t]], Im[f[t]]} in Mathematica, although it produces the same result and WA seems not to understand it. $\endgroup$ – MathIsFun7225 Dec 13 '20 at 20:26
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ReImPlot[2 t + I t^2, {t, -\[Pi], \[Pi]}]

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That is a new built-in in V12 and newer: ReImPlot

ReImPlot[{ArcSin[x], ArcCos[x]}, {x, -4, 4}, 
 PlotLabels -> "Expressions"]

enter image description here

The input that suffices for WolframAlpha is simpler

WolframAlpha["2 t+I t^2 plot -pi to pi"]

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WolframAlpha["{ArcSin[x],ArcCos[x]} plot -4 to 4"]

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has some unexpected behavior with the desired interval. It does not show the continuation.

An introductory example is

ReImPlot[{Sqrt[1 - x^2], -Sqrt[x^2 - 1]}, {x, -3, 3}]

enter image description here This separates the branches of roots very neat.

AbsArgPlot[1 + Exp[-Abs[x]] Sin[I Sin[5 x]], {x, -Pi, Pi}, 
 PlotRange -> Full]

enter image description here AbsArgPlot is an acompanying function suiting the given criteria too.

A Mathematica example is the Nyquist plot:

h = 1/(s - 1/2)^2 /. s -> Exp[I \[Omega]]
ParametricPlot[{Re[h], Im[h]}, {\[Omega], 0, 2 Pi}]

enter image description here

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