Re [z]
gives the real part of the complex number z.
Re
Re [z]
gives the real part of the complex number z.
Examples
open all close allBasic Examples (4)
Find the real part of a complex number:
Re[2 + 3I]Find the real part of a complex number expressed in polar form:
Re[2Exp[I π / 3]]Plot over a subset of the complex plane:
Plot3D[Re[x + I y], {x, -2, 2}, {y, -2, 2}]Use Re to specify regions of the complex plane:
ComplexRegionPlot[Re[z] > 0, {z, 2}]Scope (29)
Numerical Evaluation (7)
Evaluate numerically:
Re[1.2]Complex number input:
Re[(3/2) + (4/5)I]Evaluate to high precision:
N[Re[E + I Pi], 25]Mixed‐precision complex inputs:
Re[1.64 - 2I]Re[1.6400000000000000000 - 2I]Re[(164/100) - 2`20I]Re[(164/100) - 2.5I]Evaluate efficiently at high precision:
N[Re[-5^(1/(5))], 100]//TimingN[Re[-5^(1/(5))], 10000];//TimingRe threads elementwise over lists and matrices:
Re[{1.2, 1.5I, -1.8I}]Re[(| | |
| :- | :- |
| 1 | u |
| v | -I |)]Re can be used with Interval and CenteredInterval objects:
Re[Interval[{-1, 1}]]Re[CenteredInterval[-2 + 3I, 1 + 2I]]Specific Values (6)
Values of Re at fixed points:
Table[Re[n], {n, {1, I 7 / 3, -7 / 5 + I}}]Value at zero:
Re[0]Values at infinity:
Re[Infinity]Re[I Infinity]Re[DirectedInfinity[-1 + I]]Re[ComplexInfinity]Exact inputs:
Re[-5^(1/(5))]Evaluate for complex exponentials:
Table[Re[Exp[k I 3π / 8]], {k, 8}]Evaluate symbolically:
PiecewiseExpand[Re[I x], -2 < x < 2]Visualization (5)
Visualize on the real axis:
Plot[Re[x^(1/(4))], {x, -3, 3}, MaxRecursion -> 15]Plot on the real axis:
Plot[Re[E^I x], {x, -2π, 2π}]Visualize Re in the complex plane:
ComplexContourPlot[Re[z], {z, 2}, PlotLegends -> Automatic]Visualize Re in three dimensions:
ComplexPlot3D[Re[z], {z, -2 - 2I, 2 + 2I}, PlotLegends -> Automatic, MeshFunctions -> {Re[#1]&, Im[#1]&}, Mesh -> Automatic]Use Re to specify regions of the complex plane:
ComplexRegionPlot[Re[z^3] > Re[(1/z^3)], {z, 2}]//QuietFunction Properties (5)
Re is defined for all real and complex inputs:
FunctionDomain[Re[x], x]FunctionDomain[Re[x], x, Complexes]The range of Re is the whole real line:
FunctionRange[Re[x], x, y]This is true even in the complex plane:
FunctionRange[Re[x], x, y, Complexes]Re is an odd function:
Re[-x] == -Re[x]Re is not a differentiable function:
Re'[x]The difference quotient does not have a limit in the complex plane:
Underscript[, hUnderscript[ -> , ℂ]0](Re[x + h] - Re[x]/h)There is only a limit in certain directions, for example, the real direction:
Underscript[, hUnderscript[ -> , ℝ]0](Re[x + h] - Re[x]/h)Obtain this result using ComplexExpand :
ComplexExpand[Re'[x]]TraditionalForm formatting:
Re[x]//TraditionalFormFunction Identities and Simplifications (6)
Automatic simplification:
Re[{Re[z], Im[z], I Re[z], Abs[z], Conjugate[z]}]Expand assuming real variables x and y:
ComplexExpand[Re[x]]ComplexExpand[Re[x + I y]]Simplify Re using appropriate assumptions:
Simplify[Re[x], x > 0]Express a complex number as a sum of its real and imaginary parts:
FullSimplify[Re[z] + I Im[z]]Express in terms of real and imaginary parts:
Abs[x ^ 2]//FunctionExpandFind the real part of a Root expression:
Re[Root[# ^ 5 + 11# ^ 2 + 1&, 2]]//RootReduceApplications (3)
Flow around a cylinder as the real part of a complex‐valued function:
ComplexContourPlot[Re[z - (1/z)], {z, 3}, Contours -> 40, Epilog -> {Disk[{0, 0}, 1]}]Construct a bivariate real harmonic function from a complex function:
z ^ 2Sin[z + z ^ 3] + Tan[(1/z)]ComplexExpand[Re[% /. z -> x + I y]]The real part satisfies Laplace's equation:
D[%, x, x] + D[%, y, y]//SimplifyReconstruct an analytic function from its real part :
analyticReconstruct[u_, {x_, y_}, z_] := 2(u //. {x -> z / 2, y -> -I z / 2}) - (u //. {x -> 0, y -> 0})Example reconstruction:
analyticReconstruct[{x ^ 2 - y ^ 2, Exp[x]Cos[y]}, {x, y}, z]//SimplifyCheck the result:
ComplexExpand[Re[% /. z -> x + I y]]//SimplifyProperties & Relations (8)
Use Simplify and FullSimplify to simplify expressions containing Re :
{Re[Exp[I Pi / 5]x], Re[2x + I]}Simplify[%, x > 3]FullSimplify[Abs[Re[z]] ≤ Abs[z]]Prove that the disk is in the right half-plane:
FullSimplify[ForAll[z, Implies[Abs[z - 1] ≤ 1, Re[z] ≥ 0]]]ComplexExpand assumes variables to be real:
ComplexExpand[Re[(x + I y) ^ 3]]ComplexExpand[Re[Sin[x + I y]]]Here z is not assumed real, and the result should be in terms of Re and Im :
ComplexExpand[Abs[z] ^ 2, z, TargetFunctions -> {Re, Im}]FunctionExpand does not assume variables to be real:
FunctionExpand[Re[z1 + z2 + z3]]FunctionExpand[Re[(x + I y) ^ 3]]ReImPlot plots the real and imaginary parts of a function:
{Plot[{Re[ArcSin[x]], Im[ArcSin[x]]}, {x, -3, 3}],
ReImPlot[ArcSin[x], {x, -3, 3}]}Use Re to describe regions in the complex plane:
Map[ComplexRegionPlot[#, {z, 3}, PlotPoints -> 40]&, {Re[(1 + I)z - 1] < 0, -1 < Re[z ^ 2] < 1, 0 < Re[(z - 1/z + 1)] < 3, Re[z ^ 7] < 7}]Reduce can solve equations and inequalities involving Re :
Reduce[Re[z ^ 2] > 1, z]Reduce[Re[z ^ 2] == 1 && Re[1 + z] ^ 2 == Abs[z] ^ 2, z]With FindInstance you can get sample points of regions:
FindInstance[Re[z ^ 2] > 1, z]Use Re in Assumptions :
Integrate[x ^ s, {x, 1, Infinity}, Assumptions -> Re[s] < -1]Limit[1 / x ^ s, x -> Infinity, Assumptions -> Re[s] > 0]Integrate often generates conditions in terms of Re :
Integrate[x ^ s, {x, 1, Infinity}]Possible Issues (2)
Re can stay unevaluated for numeric arguments:
{Re[Log[2 + I]], Re[Sqrt[1 + I]]}Additional transformation may simplify it:
FunctionExpand[%]Re is a function of a complex variable and is therefore not differentiable:
D[Re[z], z]As a complex function, it is not possible to write Re [z] without involving Conjugate [z]:
FullSimplify[Re[z] == (z + Conjugate[z]/2)]In particular, the limit that defines the derivative is direction dependent and therefore does not exist:
Limit[DifferenceQuotient[Re[z], {z, h}], h -> 0, Direction -> 1]Limit[DifferenceQuotient[Re[z], {z, h}], h -> 0, Direction -> I]Use ComplexExpand to get differentiable expressions for real-valued variables:
D[ComplexExpand[Re[x + I y]], y]Neat Examples (1)
Use Re to plot a 3D projection of the Riemann surface of :
ParametricPlot3D[{r Cos[φ], r Sin[φ], Re[Sqrt[r]Exp[I φ / 2]]}, {r, 0, 1}, {φ, 0, 4Pi}, PlotPoints -> 40]Tech Notes
Related Links
History
Introduced in 1988 (1.0) | Updated in 2021 (13.0)
Text
Wolfram Research (1988), Re, Wolfram Language function, https://reference.wolfram.com/language/ref/Re.html (updated 2021).
CMS
Wolfram Language. 1988. "Re." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Re.html.
APA
Wolfram Language. (1988). Re. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Re.html
BibTeX
@misc{reference.wolfram_2026_re, author="Wolfram Research", title="{Re}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Re.html}", note=[Accessed: 12-June-2026]}
BibLaTeX
@online{reference.wolfram_2026_re, organization={Wolfram Research}, title={Re}, year={2021}, url={https://reference.wolfram.com/language/ref/Re.html}, note=[Accessed: 12-June-2026]}