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1.4.6 Advanced Topic: Simplifying with Assumptions
| Simplify[expr, assum] | simplify expr with assumptions |
Simplifying with assumptions. | Mathematica does not automatically simplify this, since it is only true for some values of x. | |
In[1]:=
Simplify[Sqrt[x^2]]
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| This tells Simplify to make the assumption x > 0, so that simplification can proceed. | |
In[3]:=
Simplify[Sqrt[x^2], x > 0]
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| No automatic simplification can be done on this expression. | |
In[4]:=
2 a + 2 Sqrt[a - Sqrt[-b]] Sqrt[a + Sqrt[-b]]
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If  and  are assumed to be positive, the expression can however be simplified. | |
In[5]:=
Simplify[%, a > 0 && b > 0]
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| Here is a simple example involving trigonometric functions. | |
In[6]:=
Simplify[ArcSin[Sin[x]], -Pi/2 < x < Pi/2]
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| Element[x, dom] | state that x is an element of the domain dom | Element[{ ,  , ... }, dom] | state that all the  are elements of the domain dom | | Reals | real numbers | | Integers | integers | | Primes | prime numbers |
Some domains used in assumptions. This simplifies  assuming that  is a real number. | |
In[7]:=
Simplify[Sqrt[x^2], Element[x, Reals]]
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This simplifies the sine assuming that  is an integer. | |
In[8]:=
Simplify[Sin[x + 2 n Pi], Element[n, Integers]]
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| With the assumptions given, Fermat's Little Theorem can be used. | |
In[9]:=
Simplify[Mod[a^p, p], Element[a, Integers]
&& Element[p, Primes]]
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is equal to 
for 
, but not otherwise. 
, but not 
, is real when 
is real. 
