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

Details

Examples

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Basic Examples  (4)Summary of the most common use cases

Evaluate numerically:

Out[1]=1

Use conj to conjugate expressions:

Out[1]=1

Plot over a subset of the reals:

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Plot over a subset of the complexes:

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Scope  (24)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

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Complex number input:

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Evaluate to high precision:

Out[1]=1

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

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Evaluate efficiently at high precision:

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Out[2]=2

Compute the elementwise values of an array using automatic threading:

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Or compute the matrix Conjugate function using MatrixFunction:

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Conjugate can be used with Interval and CenteredInterval objects:

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Out[2]=2

Or compute average-case statistical intervals using Around:

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Specific Values  (3)

Values of Conjugate at fixed points:

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Value at zero:

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Values at infinity:

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Out[2]=2

Visualization  (4)

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

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Plot the absolute value of function:

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

Out[1]=1

Plot the real part of function:

Out[1]=1

Plot the imaginary part of function:

Out[2]=2

Function Properties  (11)

Conjugate is defined for all real and complex inputs:

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The range of Conjugate is all real and complex values:

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Out[2]=2

Conjugate is an odd function:

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

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Conjugate is not a differentiable function:

Out[1]=1

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

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

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But in the imaginary direction, the limit is :

Out[4]=4

Conjugate is not an analytic function:

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It is singular everywhere but continuous:

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Out[3]=3

Conjugate is nondecreasing on the real line:

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Conjugate is injective on the real line:

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Out[2]=2

Conjugate is surjective on the real line:

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Out[2]=2

Conjugate is neither non-negative nor non-positive:

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TraditionalForm formatting:

Applications  (6)Sample problems that can be solved with this function

Define a scalar product for complexvalued lists utilizing BraKet notation:

Apply the definition:

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

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Recover the original fraction:

Out[3]=3

Implement a Möbius transformation:

Plot the images of concentric circles:

Out[2]=2

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

Out[2]=2

Holomorphic functions are independent of z:

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Use Conjugate to describe geometric regions:

Out[1]=1

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:

Out[2]=2

Compute the expected angular momentum in this state as :

Out[3]=3

The uncertainty in the angular momentum is :

Out[4]=4

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

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Out[6]=6

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

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Properties & Relations  (7)Properties of the function, and connections to other functions

Some transformations are performed automatically:

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Out[2]=2

Conjugate is its own inverse:

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Simplify expressions containing Conjugate:

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Out[2]=2
Out[3]=3
Out[4]=4

Assume realvalued variables:

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Out[2]=2

Assume generic complexvalued variables:

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Out[2]=2

Use Conjugate as an option value in ComplexExpand:

Out[1]=1

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

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Out[2]=2
Out[3]=3

Find Hermitian conjugate of a matrix:

Use ConjugateTranspose instead:

Possible Issues  (4)Common pitfalls and unexpected behavior

Conjugate does not always propagate into arguments:

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Differentiating Conjugate is not possible:

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

Out[2]=2
Out[3]=3

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

Out[4]=4

Conjugate can stay unevaluated for numeric arguments:

Out[1]=1
Out[2]=2

Machineprecision numeric evaluation of Conjugate can give wrong results:

Out[1]=1
Out[2]=2

Use arbitrary precision evaluation instead:

Out[3]=3
Wolfram Research (1988), Conjugate, Wolfram Language function, https://reference.wolfram.com/language/ref/Conjugate.html (updated 2021).
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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).

Copy to clipboard.
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.

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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

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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: 10-January-2025 ]}

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@misc{reference.wolfram_2024_conjugate, author="Wolfram Research", title="{Conjugate}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Conjugate.html}", note=[Accessed: 10-January-2025 ]}

BibLaTeX

@online{reference.wolfram_2024_conjugate, organization={Wolfram Research}, title={Conjugate}, year={2021}, url={https://reference.wolfram.com/language/ref/Conjugate.html}, note=[Accessed: 10-January-2025 ]}

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@online{reference.wolfram_2024_conjugate, organization={Wolfram Research}, title={Conjugate}, year={2021}, url={https://reference.wolfram.com/language/ref/Conjugate.html}, note=[Accessed: 10-January-2025 ]}