or z gives the complex conjugate of the complex number z.



open allclose all

Basic Examples  (4)

Evaluate numerically:

Use conj to conjugate expressions:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Scope  (24)

Numerical Evaluation  (6)

Evaluate numerically:

Complex number input:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

Conjugate threads elementwise over lists and matrices:

Conjugate can be used with Interval and CenteredInterval objects:

Specific Values  (3)

Values of Conjugate at fixed points:

Value at zero:

Values at infinity:

Visualization  (4)

Plot the real and imaginary parts of and over the reals:

Plot the absolute value of function:

Compare the plots of and TemplateBox[{z}, Conjugate] in three dimensions:

Plot the real part of function:

Plot the imaginary part of function:

Function Properties  (11)

Conjugate is defined for all real and complex inputs:

The range of Conjugate is all real and complex values:

Conjugate is an odd function:

Conjugate is involutive, TemplateBox[{{(, TemplateBox[{z}, Conjugate, SyntaxForm -> SuperscriptBox], )}}, Conjugate]=z:

Conjugate is not a differentiable function:

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

The limit has different values in different directions, for example, in the real direction:

But in the imaginary direction, the limit is :

Conjugate is not an analytic function:

It is singular everywhere but continuous:

Conjugate is nondecreasing on the real line:

Conjugate is injective on the real line:

Conjugate is surjective on the real line:

Conjugate is neither non-negative nor non-positive:

TraditionalForm formatting:

Applications  (6)

Define a scalar product for complexvalued lists utilizing BraKet notation:

Apply the definition:

Rewrite a complex-valued rational function into one with real denominator:

Recover the original fraction:

Implement a Möbius transformation:

Plot the images of concentric circles:

Write a realvalued function as a function of z and z:

Holomorphic functions are independent of z:

Use Conjugate to describe geometric regions:

In quantum mechanics, systems with finitely many states are represented by unit vectors and physical quantities by matrices that act on them. Consider a spin-1/2 particle such as an electron in the following state:

The operator for the component of angular momentum is given by the following matrix:

Compute the expected angular momentum in this state as :

The uncertainty in the angular momentum is :

The uncertainty in the component of angular momentum is computed analogously:

The uncertainty principle gives a lower bound on the product of uncertainties, :

Properties & Relations  (7)

Some transformations are performed automatically:

Conjugate is its own inverse:

Simplify expressions containing Conjugate:

Assume realvalued variables:

Assume generic complexvalued variables:

Use Conjugate as an option value in ComplexExpand:

Integrate along a line in the complex plane, symbolically and numerically:

Find Hermitian conjugate of a matrix:

Use ConjugateTranspose instead:

Possible Issues  (4)

Conjugate does not always propagate into arguments:

Differentiating Conjugate is not possible:

The limit that defines the derivative is direction dependent and therefore does not exist:

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

Conjugate can stay unevaluated for numeric arguments:

Machineprecision numeric evaluation of Conjugate can give wrong results:

Use arbitrary precision evaluation instead:

Wolfram Research (1988), Conjugate, Wolfram Language function, (updated 2021).


Wolfram Research (1988), Conjugate, Wolfram Language function, (updated 2021).


Wolfram Language. 1988. "Conjugate." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021.


Wolfram Language. (1988). Conjugate. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_conjugate, author="Wolfram Research", title="{Conjugate}", year="2021", howpublished="\url{}", note=[Accessed: 12-July-2024 ]}


@online{reference.wolfram_2024_conjugate, organization={Wolfram Research}, title={Conjugate}, year={2021}, url={}, note=[Accessed: 12-July-2024 ]}