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Why can't we find real and imaginary parts using Assuming?

Ernst Huijer, retired from AUB
Posted 8 months ago

Why don't I get a result based on the assumptions?

In[616]:= Assuming[{x \[Element] Reals, y \[Element] Reals}, 
 Re[x + \[ImaginaryJ] y - 1/(x + \[ImaginaryJ] y)]]

Out[616]= -Im[y] + Re[x - 1/(x + I y)]
POSTED BY: Ernst Huijer
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Yes, it looks inconsistent. The output that we would expect

x - x/(x^2 + y^2)

has the same LeafCount as

x - Re[1/(x + I y)]

so that FullSimplify probably finds them equally simple, and I don't know how the choice is done. The task of rewriting an expression in exactly the form we wish is often tricky.
POSTED BY: Gianluca Gorni

Seems like a bug or deficiency. We often want to have purely real expressions. There should be a way to force that.
POSTED BY: Ernst Huijer
POSTED BY: Gianluca Gorni
Eric Rimbey
Eric Rimbey
Posted 8 months ago
POSTED BY: Eric Rimbey

It seems that Re does not use the assumptions:

In[57]:= Assuming[{x \[Element] Reals}, Re[x]]

Out[57]= Re[x]

For you purpose you can use ComplexExpand:

ComplexExpand[Re[x + I y - 1/(x + I y)]]

I am surprised by the output of the following:

Assuming[{x \[Element] Reals, y \[Element] Reals},
 FullSimplify[ReIm[x + I y - 1/(x + I y)]]]

It may be a matter of LeafCount.
POSTED BY: Gianluca Gorni 
{x - Re[1/(x + I y)], y (1 + 1/(x^2 + y^2))}

is what I get. Not very consistent.
POSTED BY: Ernst Huijer
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