# I have a function $R \to C$, I want to plot it as the way we can plot parametric equations in the $R^2$. How to do it?

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?

• 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.
– Bill
Dec 13 '20 at 20:26
• 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. Dec 13 '20 at 20:26

ReImPlot[2 t + I t^2, {t, -\[Pi], \[Pi]}] That is a new built-in in V12 and newer: ReImPlot

ReImPlot[{ArcSin[x], ArcCos[x]}, {x, -4, 4},
PlotLabels -> "Expressions"] The input that suffices for WolframAlpha is simpler

WolframAlpha["2 t+I t^2 plot -pi to pi"] WolframAlpha["{ArcSin[x],ArcCos[x]} plot -4 to 4"] 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}] This separates the branches of roots very neat.

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