This is documentation for Mathematica 3, which was
based on an earlier version of the Wolfram Language.
 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 1.4.5, Section 3.3.8. See also: GaussianIntegers, TrigToExp, ExpToTrig, TrigExpand, FunctionExpand. Further Examples You can expand complex powers. In[1]:= Out[1]= In[2]:= Out[2]= You can expand absolute values. In[3]:= Out[3]= You can expand complex exponentials, trig functions, and hyperbolic functions. In[4]:= Out[4]= In[5]:= Out[5]= You can expand trig and hyperbolic functions of complex arguments. In[6]:= Out[6]= In[7]:= Out[7]= Using the TargetFunction optionThis is an expansion in terms of z and the absolute value of z. In[8]:= Out[8]= Now we expand in terms of polar coordinates. In[9]:= Out[9]= Finally, here is an expansion in terms of z and its conjugate. In[10]:= Out[10]=