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Conjugate
✖
Conjugate

✖

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

Details

Examples

open allclose all

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

Evaluate numerically:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-bvansf
Out[1]=1

Use conj to conjugate expressions:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-bbwnbc
Out[1]=1

Plot over a subset of the reals:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-6en3s
Out[1]=1

Plot over a subset of the complexes:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-p3ah
Out[1]=1

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

Numerical Evaluation  (6)

Evaluate numerically:

In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-cksbl4
Out[2]=2

Complex number input:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-l274ju
Out[1]=1

Evaluate to high precision:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-b0wt9
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-y7k4a
Out[2]=2

Evaluate efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-di5gcr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-bq2c6r
Out[2]=2

Compute the elementwise values of an array using automatic threading:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-thgd2
Out[1]=1

Or compute the matrix Conjugate function using MatrixFunction:

In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-o5jpo
Out[2]=2

Conjugate can be used with Interval and CenteredInterval objects:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-bwmui
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-k7pzcv
Out[2]=2

Or compute average-case statistical intervals using Around:

In[3]:=3
✖
https://wolfram.com/xid/0bn5y9u-cw18bq
Out[3]=3

Specific Values  (3)

Values of Conjugate at fixed points:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-nww7l
Out[1]=1

Value at zero:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-bmqd0y
Out[1]=1

Values at infinity:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-b68yf7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-cuau10
Out[2]=2

Visualization  (4)

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

In[3]:=3
✖
https://wolfram.com/xid/0bn5y9u-0ex7ly
Out[3]=3

Plot the absolute value of function:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-eq46t4
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-i75zi3
Out[1]=1

Plot the real part of function:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-kgd8nu
Out[1]=1

Plot the imaginary part of function:

In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-dejvir
Out[2]=2

Function Properties  (11)

Conjugate is defined for all real and complex inputs:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-cl7ele
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-c4ycek
Out[2]=2

The range of Conjugate is all real and complex values:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-evf2yr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-hi2o8d
Out[2]=2

Conjugate is an odd function:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-s5zo8
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-f8zeh
Out[1]=1

Conjugate is not a differentiable function:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-fb9jdx
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-fqx7yy
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bn5y9u-yvnsee
Out[3]=3

But in the imaginary direction, the limit is :

In[4]:=4
✖
https://wolfram.com/xid/0bn5y9u-cfnpct
Out[4]=4

Conjugate is not an analytic function:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-h5x4l2
Out[1]=1

It is singular everywhere but continuous:

In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-mdtl3h
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn5y9u-mn5jws
Out[3]=3

Conjugate is nondecreasing on the real line:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-nlz7s
Out[1]=1

Conjugate is injective on the real line:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-poz8g
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-ctca0g
Out[2]=2

Conjugate is surjective on the real line:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-cxk3a6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-frlnsr
Out[2]=2

Conjugate is neither non-negative nor non-positive:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-84dui
Out[1]=1

TraditionalForm formatting:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-b5n6p1

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

Define a scalar product for complex‐valued lists utilizing BraKet notation:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-jszvlo

Apply the definition:

In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-ihss59
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-bhe6l4
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-e01ub
Out[2]=2

Recover the original fraction:

In[3]:=3
✖
https://wolfram.com/xid/0bn5y9u-fxvdp6
Out[3]=3

Implement a Möbius transformation:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-gdtf05

Plot the images of concentric circles:

In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-hkltw
Out[2]=2

Write a real‐valued function as a function of z and z:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-bn0cwr
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-gmjox
Out[2]=2

Holomorphic functions are independent of z:

In[3]:=3
✖
https://wolfram.com/xid/0bn5y9u-e5gutj
Out[3]=3

Use Conjugate to describe geometric regions:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-ii3cw3
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:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-38x16q

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

In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-sja8mn
Out[2]=2

Compute the expected angular momentum in this state as :

In[3]:=3
✖
https://wolfram.com/xid/0bn5y9u-3srgxm
Out[3]=3

The uncertainty in the angular momentum is :

In[4]:=4
✖
https://wolfram.com/xid/0bn5y9u-xtk4yx
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0bn5y9u-l30enu
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0bn5y9u-9r9je9
Out[6]=6

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

In[7]:=7
✖
https://wolfram.com/xid/0bn5y9u-kcsb44
Out[7]=7

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

Some transformations are performed automatically:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-h005bq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-dzzha4
Out[2]=2

Conjugate is its own inverse:

In[3]:=3
✖
https://wolfram.com/xid/0bn5y9u-dp0ll4
Out[3]=3

Simplify expressions containing Conjugate:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-in74ys
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-khwkq
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn5y9u-ci3w53
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bn5y9u-bxsdqk
Out[4]=4

Assume real‐valued variables:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-m5epk1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-ipi3kg
Out[2]=2

Assume generic complex‐valued variables:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-gptum8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-56raw
Out[2]=2

Use Conjugate as an option value in ComplexExpand:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-h9w2a2
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-nj0qd3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-exluyb
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn5y9u-hiyuc3
Out[3]=3

Find Hermitian conjugate of a matrix:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-bqnc20
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-klmvj2

Use ConjugateTranspose instead:

In[3]:=3
✖
https://wolfram.com/xid/0bn5y9u-brjrsy

Possible Issues  (4)Common pitfalls and unexpected behavior

Conjugate does not always propagate into arguments:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-kpb9av
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-gc4tpd
Out[2]=2

Differentiating Conjugate is not possible:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-c65v77
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-2pb7j7
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bn5y9u-xvhk4r
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0bn5y9u-zvk3va
Out[4]=4

Conjugate can stay unevaluated for numeric arguments:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-itqxfr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-h2gll8
Out[2]=2

Machine‐precision numeric evaluation of Conjugate can give wrong results:

In[1]:=1
✖
https://wolfram.com/xid/0bn5y9u-okperw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bn5y9u-iobx5c
Out[2]=2

Use arbitrary precision evaluation instead:

In[3]:=3
✖
https://wolfram.com/xid/0bn5y9u-bka6r7
Out[3]=3
Wolfram Research (1988), Conjugate, Wolfram Language function, https://reference.wolfram.com/language/ref/Conjugate.html (updated 2021).
✖
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).

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_conjugate, author="Wolfram Research", title="{Conjugate}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Conjugate.html}", note=[Accessed: 29-March-2025 ]}

✖
@misc{reference.wolfram_2025_conjugate, author="Wolfram Research", title="{Conjugate}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Conjugate.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_conjugate, organization={Wolfram Research}, title={Conjugate}, year={2021}, url={https://reference.wolfram.com/language/ref/Conjugate.html}, note=[Accessed: 29-March-2025 ]}

✖
@online{reference.wolfram_2025_conjugate, organization={Wolfram Research}, title={Conjugate}, year={2021}, url={https://reference.wolfram.com/language/ref/Conjugate.html}, note=[Accessed: 29-March-2025 ]}