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Problem with simplifying expression containing Conjugate[]

Tigran Aivazian, Bibles.org.uk
Posted 3 years ago

The following expression should simplify to 1, but instead it is returned unevaluated:

Mathematica 12.1.0 Kernel for Linux x86 (64-bit)
Copyright 1988-2020 Wolfram Research, Inc.

In[1]:= $Assumptions = a>0 && b>0

Out[1]= a > 0 && b > 0

In[2]:= Sqrt[a+I*b]*Conjugate[1/Sqrt[a-I*b]] // FullSimplify

                                      1
Out[2]= Sqrt[a + I b] Conjugate[-------------]
                                Sqrt[a - I b]

Note that the equivalent code in Maxima evaluates to 1 as expected:

(%i2) assume(a>0,b>0);
(%o2)                           [a > 0, b > 0]
(%i3) sqrt(a+%i*b)*conjugate(1/sqrt(a-%i*b));
(%o3)                                  1
POSTED BY: Tigran Aivazian
2 Replies
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Thank you, yes, both versions work fine:

In[4]:= Sqrt[a+I*b]*Conjugate[1/Sqrt[a-I*b]] // ComplexExpand[#, TargetFunctions->Conjugate] & // Simplify[#, Assumptions->{a>0,b>0}] &

Out[4]= 1

In[5]:= Sqrt[a+I*b]*Conjugate[1/Sqrt[a-I*b]] // ComplexExpand[#, TargetFunctions->{Re,Im}] & // Simplify[#, Assumptions->{a>0,b>0}] &

Out[5]= 1
POSTED BY: Tigran Aivazian

Try:

Simplify[ComplexExpand[Sqrt[a + I*b]*Conjugate[1/Sqrt[a - I*b]], 
  TargetFunctions -> {Re, Im}], Assumptions -> {a > 0, b > 0}]

Or:

Simplify[ComplexExpand[Sqrt[a + I*b]*Conjugate[1/Sqrt[a - I*b]], 
  TargetFunctions -> Conjugate], Assumptions -> {a > 0, b > 0}]
POSTED BY: Mariusz Iwaniuk
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