# 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:

Mixedprecision 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:

### 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 complexvalued 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:

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: 19-September-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: 19-September-2024 ]}