Evaluate numerically:

Use conj to conjugate expressions:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Define a scalar product for complex‐valued lists utilizing BraKet notation:

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:

In quantum mechanics, systems with finitely many states are represented by unit vectors and physical quantities by matrices that act on them. Consider a spin-1/2 particle such as an electron in the following state:

The operator for the component of angular momentum is given by the following matrix:

Compute the expected angular momentum in this state as :

The uncertainty in the angular momentum is :

The uncertainty in the component of angular momentum is computed analogously:

The uncertainty principle gives a lower bound on the product of uncertainties, :

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:

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:

Use arbitrary precision evaluation instead: