# Conjugate

Conjugate[z]

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

# Examples

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## 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 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, :

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:

## 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:

## 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:

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

#### Text

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

#### CMS

Wolfram Language. 1988. "Conjugate." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Conjugate.html.

#### APA

Wolfram Language. (1988). Conjugate. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Conjugate.html

#### BibTeX

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

#### BibLaTeX

@online{reference.wolfram_2024_conjugate, organization={Wolfram Research}, title={Conjugate}, year={2021}, url={https://reference.wolfram.com/language/ref/Conjugate.html}, note=[Accessed: 12-July-2024 ]}