Re

Re[z]

gives the real part of the complex number z.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Re[expr] is left unevaluated if expr is not a numeric quantity.
  • Re automatically threads over lists.
  • Re can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (4)

Find the real part of a complex number:

Find the real part of a complex number expressed in polar form:

Plot over a subset of the complex plane:

Use Re to specify regions of the complex plane:

Scope  (29)

Numerical Evaluation  (7)

Evaluate numerically:

Complex number input:

Evaluate to high precision:

Mixed‐precision complex inputs:

Evaluate efficiently at high precision:

Re threads elementwise over lists and matrices:

Re can be used with Interval and CenteredInterval objects:

Specific Values  (6)

Values of Re at fixed points:

Value at zero:

Values at infinity:

Exact inputs:

Evaluate for complex exponentials:

Evaluate symbolically:

Visualization  (5)

Visualize on the real axis:

Plot on the real axis:

Visualize Re in the complex plane:

Visualize Re in three dimensions:

Use Re to specify regions of the complex plane:

Function Properties  (5)

Re is defined for all real and complex inputs:

The range of Re is the whole real line:

This is true even in the complex plane:

Re is an odd function:

Re is not a differentiable function:

The difference quotient does not have a limit in the complex plane:

There is only a limit in certain directions, for example, the real direction:

Obtain this result using ComplexExpand:

TraditionalForm formatting:

Function Identities and Simplifications  (6)

Automatic simplification:

Expand assuming real variables x and y:

Simplify Re using appropriate assumptions:

Express a complex number as a sum of its real and imaginary parts:

Express in terms of real and imaginary parts:

Find the real part of a Root expression:

Applications  (3)

Flow around a cylinder as the real part of a complex‐valued function:

Construct a bivariate real harmonic function from a complex function:

The real part satisfies Laplace's equation:

Reconstruct an analytic function from its real part :

Example reconstruction:

Check the result:

Properties & Relations  (8)

Use Simplify and FullSimplify to simplify expressions containing Re:

Prove that the disk is in the right half-plane:

ComplexExpand assumes variables to be real:

Here z is not assumed real, and the result should be in terms of Re and Im:

FunctionExpand does not assume variables to be real:

ReImPlot plots the real and imaginary parts of a function:

Use Re to describe regions in the complex plane:

Reduce can solve equations and inequalities involving Re:

With FindInstance you can get sample points of regions:

Use Re in Assumptions:

Integrate often generates conditions in terms of Re:

Possible Issues  (2)

Re can stay unevaluated for numeric arguments:

Additional transformation may simplify it:

Re is a function of a complex variable and is therefore not differentiable:

As a complex function, it is not possible to write Re[z] without involving Conjugate[z]:

In particular, the limit that defines the derivative is direction dependent and therefore does not exist:

Use ComplexExpand to get differentiable expressions for real-valued variables:

Neat Examples  (1)

Use Re to plot a 3D projection of the Riemann surface of :

Wolfram Research (1988), Re, Wolfram Language function, https://reference.wolfram.com/language/ref/Re.html (updated 2021).

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_2024_re, author="Wolfram Research", title="{Re}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Re.html}", note=[Accessed: 22-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_re, organization={Wolfram Research}, title={Re}, year={2021}, url={https://reference.wolfram.com/language/ref/Re.html}, note=[Accessed: 22-December-2024 ]}