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Conjugate

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Conjugate[z] gives the complex conjugate Re[z] - I Im[z] of numeric z; real and real-valued (Re/Im/Abs/Arg) arguments are returned unchanged.

Examples (5)

Every input below was run against the current Mathilda build and its output recorded.

Applications (5)

In[1]:= Conjugate[3 + 4 I]
Out[1]= 3 - 4*I

In[2]:= Conjugate[5]
Out[2]= 5

In[3]:= Conjugate[{1 + I, 2 - 3 I}]
Out[3]= {1 - I, 2 + 3*I}

In[4]:= Conjugate[(2 + I)/(1 - 3 I)]
Out[4]= -1/10 - 7/10*I

In[5]:= z Conjugate[z] /. z -> 3 + 4 I
Out[5]= 25

Implementation notes

builtin_conjugate folds the involution Conjugate[Conjugate[z]] -> z and treats the real-valued-by-construction heads Re, Im, Abs, Arg as fixed points. For a Complex[re, im] literal it returns make_complex(re, -im); for real numerics (Integer/Real/Rational) and any expression that is_numeric_real (e.g. Sqrt[2], Pi) it returns the argument unchanged. An expression that complex_decompose splits into concretely-numeric real/imag parts is conjugated as re - im*I. Symbolic inputs return NULL (a one-argument arity check emits Conjugate::argx).

Attributes: Listable, NumericFunction, Protected.

References

See also: Re, Im, ReIm, Abs, Sign, Arg, Rational, Complex

Notes & additional examples

Notes

Conjugate[z] returns Re[z] - I Im[z]; real arguments are returned unchanged. It is Listable.