ComplexExpand, Conjugate, Piecewise - an unexpected result

Questions
,
1 
ComplexExpand[Conjugate[Piecewise[{{-I,a>0}},I]]]

The above expresstion yields (a is an undetermined symbol, you can replace it with 1 if you like)

Piecewise[{{-I,a>0}},I]

I expect this expression to yield

Piecewise[{{I,a>0}},-I]

because this is what conjugate do. It seems that Conjugate function have been ignored.

2

The following code produce the same thing.

In[3]:= Conjugate[f[x]]
Out[3]= Conjugate[f[x]]
In[4]:= ComplexExpand[%]
Out[4]= f[x]

The documentation says it does not always propagate into the expression. Once it stays outside, the ComplexExpand simply kills the head.

3

I still don’t understand.

The document just say it does not always propagate into arguments, but how can ComplexExpand kill it? This gives wrong results!

During a very complex calculation, I just cannot make sure every Conjugate is well dealt with (they are just intermediate results that I cannot see), and it accually took me hours to find this problem in a very big calculation.

4

I agree. It should stay like the input if it is not sure.

ComplexExpand[Conjugate[Piecewise[{{-I, a > 0}}, I]]]

If you give “a” a value the result is correct

In[31]:= ComplexExpand[Conjugate[Piecewise[{{-I,1>0}},I]]]
Out[31]= I

or use set delay

In[32]:= f[a_]:=ComplexExpand[Conjugate[Piecewise[{{-I,a>0}},I]]]
In[33]:= f[1]
Out[33]= I

You can make a suggestion to support@wolfram.com about this concern.

5

Conjugate is primarily for numbers and expressions that represent numbers (such as Root objects).

If you want Conjugate to work on the parameters inside an expression,
use Map with a level specification. For example,

In[1]:= Map[ Conjugate, Piecewise[{{-I, a > 0}}, I], {-1} ]
Out[1]= Piecewise[{{I, Conjugate[a] > 0}}, -I]

In[2]:= FullSimplify[ %,  Element[a, Reals] ]
Out[2]= Piecewise[{{-I, a <= 0}}, I]