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ComplexExpand

ComplexExpand[expr]
expands expr assuming that all variables are real.

ComplexExpand[expr, {x₁, x₂, ...}]
expands expr assuming that variables matching any of the x_i are complex.

The option TargetFunctions can be given as a list of functions from the set {Re, Im, Abs, Arg, Conjugate, Sign}. ComplexExpand will try to give results in terms of functions specified.

ComplexExpand[expr, vars, TargetFunctions->{Abs, Arg}] converts to polar coordinates.

ComplexExpand automatically threads over lists in expr, as well as equations, inequalities and logic functions.

Assume both x and y are real:

Take x to be complex:

Assume both x and y are real:

Take x to be complex:

Polynomials:

Trigonometric and hyperbolic functions:

Inverse trigonometric and inverse hyperbolic functions:

Exponential and logarithmic functions:

Composition of functions:

This gives an answer in terms of Re[z] and Im[z]:

With TargetFunctions->{Abs, Arg}, the answer is given in terms of Abs[z] and Arg[z]:

Use Conjugate as the target function:

This computes Re[Sin[x+I y]] assuming that x and y are real:

The same computation can be done using TrigExpand and Refine:

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