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or z* gives the complex conjugate of the complex number z.
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • can be entered as Esc co Esc, Esc conj Esc, or \[Conjugate].
Use Esc conj Esc to conjugate expressions:
Click for copyable input
Use Esc conj Esc to conjugate expressions:
Click for copyable input
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:
Infinite arguments give symbolic results:
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:
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:
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:
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