ComplexExpand

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`ComplexExpand`

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ComplexExpand[expr]

expands expr assuming that all variables are real.

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ComplexExpand[expr,{x₁,x₂,…}]

expands expr assuming that variables matching any of the x_i are complex.

Details and Options

The variables given in the second argument of ComplexExpand can be patterns.
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 automatically threads over lists in expr, as well as equations, inequalities and logic functions.

Examples

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Basic Examples (4)Summary of the most common use cases

Expand symbolic expressions into real and imaginary parts:

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-gk1xuc

Out[1]=1

Assume that both and are real:

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-837p3e

Out[1]=1

Take to be complex:

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-dk9pdd

Out[1]=1

Extract the real and imaginary parts of an expression:

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-292n1e

Out[1]=1

Scope (7)Survey of the scope of standard use cases

Polynomials:

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-mzchue

Out[1]=1

Trigonometric and hyperbolic functions:

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-h17pcy

Out[1]=1

Inverse trigonometric and inverse hyperbolic functions:

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-deeg6d

Out[1]=1

Exponential and logarithmic functions:

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-cur2us

Out[1]=1

Composition of functions:

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-ekcug4

Out[1]=1

Specify that a variable is taken to be complex:

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-jt2sm8

Out[1]=1

Specify target functions:

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-hadptl

Out[1]=1

Options (1)Common values & functionality for each option

TargetFunctions (1)

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

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-ekv4lp

Out[1]=1

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

In[2]:=2

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https://wolfram.com/xid/0i1kvjxlu-i6i3wv

Out[2]=2

Use Conjugate as the target function:

In[3]:=3

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https://wolfram.com/xid/0i1kvjxlu-ek2gaj

Out[3]=3

Applications (2)Sample problems that can be solved with this function

This expands the expression, assuming that and are both real:

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-wzj

Out[1]=1

In this case, is assumed to be real, but is assumed to be complex, and is broken into explicit real and imaginary parts:

In[2]:=2

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https://wolfram.com/xid/0i1kvjxlu-jkj

Out[2]=2

With several complex variables, you quickly get quite complicated results:

In[3]:=3

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https://wolfram.com/xid/0i1kvjxlu-ttv

Out[3]=3

Verify common complex identities:

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-b56gd9

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0i1kvjxlu-m6badi

Out[2]=2

Properties & Relations (1)Properties of the function, and connections to other functions

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

In[1]:=1

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https://wolfram.com/xid/0i1kvjxlu-bpkhhf

Out[1]=1

The same computation can be done using TrigExpand and Refine:

In[2]:=2

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https://wolfram.com/xid/0i1kvjxlu-b67d3t

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0i1kvjxlu-jmdat

Out[3]=3

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