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

and

:

Holomorphic functions are independent of

:

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: