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ComplexExpand

  • ComplexExpand[ expr ] expands expr assuming that all variables are real.
  • ComplexExpand[ expr , , , ... ] expands expr assuming that variables matching any of the are complex.
  • Example: ComplexExpand[Sin[x + I y]].
  • The variables given in the second argument of ComplexExpand can be patterns.
  • Example: ComplexExpand[Sin[x], x].
  • 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.
  • See the Mathematica book: Section 1.4.5Section 3.3.8.
  • See also: GaussianIntegers, TrigToExp, ExpToTrig, TrigExpand, FunctionExpand.

    Further Examples

    You can expand complex powers.

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    You can expand absolute values.

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    You can expand complex exponentials, trig functions, and hyperbolic functions.

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    You can expand trig and hyperbolic functions of complex arguments.

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    Using the TargetFunction option
    This is an expansion in terms of z and the absolute value of z.

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    Now we expand in terms of polar coordinates.

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    Finally, here is an expansion in terms of z and its conjugate.

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