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Failure to fully incorporate assumptions (or am I missing/doing something )

David S

Posted 11 years ago

Input of: Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b}, Assumptions -> -1 < a && a < 0 && b > a && b < +1] Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b}, Assumptions -> -1 < a && a > 0 && b > a && b < +1] Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b}, Assumptions -> -1 < a && b > a && b < +1] gives output: Sqrt[1 - a^2] - Sqrt[1 - b^2] Sqrt[1 - a^2] - Sqrt[1 - b^2] ConditionalExpression[Sqrt[1 - a^2] - Sqrt[1 - b^2], a > 0] Inputs 1. and 2. give outputs 1. and 2. that are correct (and also true/correct for a=0). The third output is correct, but not as general a statement as I would expect. The first and second input equations are identical except their assumptions of a < 0 and a > 0, respectively. The third skips that part of the assumptions totally. I would have expected/hoped that input 3's output would be identical to 1 and 2's outputs. What gives? Am I doing something wrong or missing something or is this just a shortcoming of M8? Thanks, ds

POSTED BY: David S

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Udo Krause

Udo Krause, NOrganization

Posted 11 years ago

If that was in Mathematica 8 (M8) ... it's still here in Mathematica 10.0.1 In[1]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b}, Assumptions -> -1 < a && a < 0 && b > a && b < +1] Out[1]= Sqrt[1 - a^2] - Sqrt[1 - b^2] In[2]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b}, Assumptions -> -1 < a && a > 0 && b > a && b < +1] Out[2]= Sqrt[1 - a^2] - Sqrt[1 - b^2] In[3]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b}, Assumptions -> -1 < a && b > a && b < +1] Out[3]= ConditionalExpression[Sqrt[1 - a^2] - Sqrt[1 - b^2], a > 0] In[4]:= $Version Out[4]= "10.0 for Microsoft Windows (64-bit) (September 9, 2014)" In[5]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b}, Assumptions -> -1 < a < b < 1] Out[5]= ConditionalExpression[Sqrt[1 - a^2] - Sqrt[1 - b^2], a > 0] In[6]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b}, Assumptions -> -1 < a && ((a < 0) \|\| (a > 0)) && a < b && b < 1] Out[6]= ConditionalExpression[Sqrt[1 - a^2] - Sqrt[1 - b^2], a > 0] In[8]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b}, Assumptions -> -1 < a && a != 0 && a < b && b < 1] Out[8]= ConditionalExpression[Sqrt[1 - a^2] - Sqrt[1 - b^2], a > 0]

If that was in Mathematica 8 (M8) ... it's still here in Mathematica 10.0.1

In[1]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b}, Assumptions -> -1 < a && a < 0 && b > a && b < +1]
Out[1]= Sqrt[1 - a^2] - Sqrt[1 - b^2]

In[2]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b},  Assumptions -> -1 < a && a > 0 && b > a && b < +1]
Out[2]= Sqrt[1 - a^2] - Sqrt[1 - b^2]

In[3]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b},  Assumptions -> -1 < a && b > a && b < +1]
Out[3]= ConditionalExpression[Sqrt[1 - a^2] - Sqrt[1 - b^2], a > 0]

In[4]:= $Version
Out[4]= "10.0 for Microsoft Windows (64-bit) (September 9, 2014)"

In[5]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b},  Assumptions -> -1 < a < b < 1]
Out[5]= ConditionalExpression[Sqrt[1 - a^2] - Sqrt[1 - b^2], a > 0]

In[6]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b},  Assumptions -> -1 < a && ((a < 0) || (a > 0)) && a < b && b < 1]
Out[6]= ConditionalExpression[Sqrt[1 - a^2] - Sqrt[1 - b^2], a > 0]

In[8]:= Integrate[ChebyshevT[1, z] (1 - z^2)^(-1/2), {z, a, b}, Assumptions -> -1 < a && a != 0 && a < b && b < 1]
Out[8]= ConditionalExpression[Sqrt[1 - a^2] - Sqrt[1 - b^2], a > 0]

POSTED BY: Udo Krause

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