Built-in Mathematica Symbol

Conjugate

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

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

* can be entered as Esc co Esc, Esc conj Esc or \[Conjugate].

Conjugate automatically threads over lists.

EXAMPLES

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Basic Examples (2)

Use Esc conj Esc to conjugate expressions:

In[1]:=

Out[1]=

Use Esc conj Esc to conjugate expressions:

In[1]:=

Out[1]=

Scope (5)

Find complex conjugate of complex exponentials:

Find complex conjugate of explicit algebraic numbers:

The precision of the output does not track the precision of the input:

Conjugate threads element-wise over lists:

TraditionalForm formatting:

Generalizations & Extensions (1)

Infinite arguments give symbolic results:

Applications (5)

Define a scalar product for complex-valued lists:

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 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 real-valued variables:

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:

Conjugate can stay unevaluated for numeric arguments:

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

Use arbitrary precision evaluation instead:

Differentiating Conjugate is not possible: