Conjugate

Conjugate[z]
or z gives the complex conjugate of the complex number z.
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- can be entered as
co
,
conj
, or \[Conjugate].
- Conjugate can be used with Interval and CenteredInterval objects. »
- Conjugate automatically threads over lists.
Examples
open allclose allBasic Examples (4)
Scope (24)
Numerical Evaluation (6)
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)
Visualization (4)
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:
TraditionalForm formatting:
Applications (5)
Define a scalar product for complex‐valued lists:
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 real‐valued function as a function of z and z:
Holomorphic functions are independent of z:
Use Conjugate to describe geometric regions:
Properties & Relations (7)
Some transformations are performed automatically:
Conjugate is its own inverse:
Simplify expressions containing Conjugate:
Assume generic complex‐valued 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:


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