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Conjugate
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- can be entered as co, conj, or \[Conjugate].
- Conjugate automatically threads over lists. »
- Conjugate can be used with Interval and CenteredInterval objects. »
Examples
open allclose allBasic Examples (4)Summary of the most common use cases
https://wolfram.com/xid/0bn5y9u-bvansf
Use conj to conjugate expressions:
https://wolfram.com/xid/0bn5y9u-bbwnbc
Plot over a subset of the reals:
https://wolfram.com/xid/0bn5y9u-6en3s
Plot over a subset of the complexes:
https://wolfram.com/xid/0bn5y9u-p3ah
Scope (24)Survey of the scope of standard use cases
Numerical Evaluation (6)
https://wolfram.com/xid/0bn5y9u-cksbl4
https://wolfram.com/xid/0bn5y9u-l274ju
https://wolfram.com/xid/0bn5y9u-b0wt9
The precision of the output tracks the precision of the input:
https://wolfram.com/xid/0bn5y9u-y7k4a
Evaluate efficiently at high precision:
https://wolfram.com/xid/0bn5y9u-di5gcr
https://wolfram.com/xid/0bn5y9u-bq2c6r
Compute the elementwise values of an array using automatic threading:
https://wolfram.com/xid/0bn5y9u-thgd2
Or compute the matrix Conjugate function using MatrixFunction:
https://wolfram.com/xid/0bn5y9u-o5jpo
Conjugate can be used with Interval and CenteredInterval objects:
https://wolfram.com/xid/0bn5y9u-bwmui
https://wolfram.com/xid/0bn5y9u-k7pzcv
Or compute average-case statistical intervals using Around:
https://wolfram.com/xid/0bn5y9u-cw18bq
Specific Values (3)
Values of Conjugate at fixed points:
https://wolfram.com/xid/0bn5y9u-nww7l
https://wolfram.com/xid/0bn5y9u-bmqd0y
https://wolfram.com/xid/0bn5y9u-b68yf7
https://wolfram.com/xid/0bn5y9u-cuau10
Visualization (4)
Plot the real and imaginary parts of and over the reals:
https://wolfram.com/xid/0bn5y9u-0ex7ly
Plot the absolute value of function:
https://wolfram.com/xid/0bn5y9u-eq46t4
Compare the plots of and in three dimensions:
https://wolfram.com/xid/0bn5y9u-i75zi3
Plot the real part of function:
https://wolfram.com/xid/0bn5y9u-kgd8nu
Plot the imaginary part of function:
https://wolfram.com/xid/0bn5y9u-dejvir
Function Properties (11)
Conjugate is defined for all real and complex inputs:
https://wolfram.com/xid/0bn5y9u-cl7ele
https://wolfram.com/xid/0bn5y9u-c4ycek
The range of Conjugate is all real and complex values:
https://wolfram.com/xid/0bn5y9u-evf2yr
https://wolfram.com/xid/0bn5y9u-hi2o8d
Conjugate is an odd function:
https://wolfram.com/xid/0bn5y9u-s5zo8
Conjugate is involutive, :
https://wolfram.com/xid/0bn5y9u-f8zeh
Conjugate is not a differentiable function:
https://wolfram.com/xid/0bn5y9u-fb9jdx
The difference quotient does not have a limit in the complex plane:
https://wolfram.com/xid/0bn5y9u-fqx7yy
The limit has different values in different directions, for example, in the real direction:
https://wolfram.com/xid/0bn5y9u-yvnsee
But in the imaginary direction, the limit is :
https://wolfram.com/xid/0bn5y9u-cfnpct
Conjugate is not an analytic function:
https://wolfram.com/xid/0bn5y9u-h5x4l2
It is singular everywhere but continuous:
https://wolfram.com/xid/0bn5y9u-mdtl3h
https://wolfram.com/xid/0bn5y9u-mn5jws
Conjugate is nondecreasing on the real line:
https://wolfram.com/xid/0bn5y9u-nlz7s
Conjugate is injective on the real line:
https://wolfram.com/xid/0bn5y9u-poz8g
https://wolfram.com/xid/0bn5y9u-ctca0g
Conjugate is surjective on the real line:
https://wolfram.com/xid/0bn5y9u-cxk3a6
https://wolfram.com/xid/0bn5y9u-frlnsr
Conjugate is neither non-negative nor non-positive:
https://wolfram.com/xid/0bn5y9u-84dui
TraditionalForm formatting:
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:
https://wolfram.com/xid/0bn5y9u-jszvlo
https://wolfram.com/xid/0bn5y9u-ihss59
Rewrite a complex-valued rational function into one with real denominator:
https://wolfram.com/xid/0bn5y9u-bhe6l4
https://wolfram.com/xid/0bn5y9u-e01ub
Recover the original fraction:
https://wolfram.com/xid/0bn5y9u-fxvdp6
Implement a Möbius transformation:
https://wolfram.com/xid/0bn5y9u-gdtf05
Plot the images of concentric circles:
https://wolfram.com/xid/0bn5y9u-hkltw
Write a real‐valued function as a function of z and z:
https://wolfram.com/xid/0bn5y9u-bn0cwr
https://wolfram.com/xid/0bn5y9u-gmjox
Holomorphic functions are independent of z:
https://wolfram.com/xid/0bn5y9u-e5gutj
Use Conjugate to describe geometric regions:
https://wolfram.com/xid/0bn5y9u-ii3cw3
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:
https://wolfram.com/xid/0bn5y9u-38x16q
The operator for the component of angular momentum is given by the following matrix:
https://wolfram.com/xid/0bn5y9u-sja8mn
Compute the expected angular momentum in this state as :
https://wolfram.com/xid/0bn5y9u-3srgxm
The uncertainty in the angular momentum is :
https://wolfram.com/xid/0bn5y9u-xtk4yx
The uncertainty in the component of angular momentum is computed analogously:
https://wolfram.com/xid/0bn5y9u-l30enu
https://wolfram.com/xid/0bn5y9u-9r9je9
The uncertainty principle gives a lower bound on the product of uncertainties, :
https://wolfram.com/xid/0bn5y9u-kcsb44
Properties & Relations (7)Properties of the function, and connections to other functions
Some transformations are performed automatically:
https://wolfram.com/xid/0bn5y9u-h005bq
https://wolfram.com/xid/0bn5y9u-dzzha4
Conjugate is its own inverse:
https://wolfram.com/xid/0bn5y9u-dp0ll4
Simplify expressions containing Conjugate:
https://wolfram.com/xid/0bn5y9u-in74ys
https://wolfram.com/xid/0bn5y9u-khwkq
https://wolfram.com/xid/0bn5y9u-ci3w53
https://wolfram.com/xid/0bn5y9u-bxsdqk
https://wolfram.com/xid/0bn5y9u-m5epk1
https://wolfram.com/xid/0bn5y9u-ipi3kg
Assume generic complex‐valued variables:
https://wolfram.com/xid/0bn5y9u-gptum8
https://wolfram.com/xid/0bn5y9u-56raw
Use Conjugate as an option value in ComplexExpand:
https://wolfram.com/xid/0bn5y9u-h9w2a2
Integrate along a line in the complex plane, symbolically and numerically:
https://wolfram.com/xid/0bn5y9u-nj0qd3
https://wolfram.com/xid/0bn5y9u-exluyb
https://wolfram.com/xid/0bn5y9u-hiyuc3
Find Hermitian conjugate of a matrix:
https://wolfram.com/xid/0bn5y9u-bqnc20
https://wolfram.com/xid/0bn5y9u-klmvj2
Use ConjugateTranspose instead:
https://wolfram.com/xid/0bn5y9u-brjrsy
Possible Issues (4)Common pitfalls and unexpected behavior
Conjugate does not always propagate into arguments:
https://wolfram.com/xid/0bn5y9u-kpb9av
https://wolfram.com/xid/0bn5y9u-gc4tpd
Differentiating Conjugate is not possible:
https://wolfram.com/xid/0bn5y9u-c65v77
The limit that defines the derivative is direction dependent and therefore does not exist:
https://wolfram.com/xid/0bn5y9u-2pb7j7
https://wolfram.com/xid/0bn5y9u-xvhk4r
Use ComplexExpand to get differentiable expressions for real-valued variables:
https://wolfram.com/xid/0bn5y9u-zvk3va
Conjugate can stay unevaluated for numeric arguments:
https://wolfram.com/xid/0bn5y9u-itqxfr
https://wolfram.com/xid/0bn5y9u-h2gll8
Machine‐precision numeric evaluation of Conjugate can give wrong results:
https://wolfram.com/xid/0bn5y9u-okperw
https://wolfram.com/xid/0bn5y9u-iobx5c
Use arbitrary precision evaluation instead:
https://wolfram.com/xid/0bn5y9u-bka6r7
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_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
]}