Wider Support For Domain Specifications
Mathematica 5 offers enhanced support for domain specifications
in many symbolic computations. The new function Refine gives the form of an expression
that would be obtained if symbols in it were replaced by explicit
numerical expressions satisfying the assumptions. The new function Assuming lets users specify one set of
assumptions to use in a whole collection of operations.
Example: Simplifying a Square Root
The following refines e, assuming that x and y
are positive.
![Refine[e, x > 0 [And] y > 0]](images/assuming_5.gif "Refine[e, x > 0 [And] y > 0]")
Example: Using Assumptions in Mathematica Functions
Assumptions are taken into account by many Mathematica
functions, and propagate through multiple nesting--often simplifying
the answer considerably or even making a close-form solution
possible at all.
![F = ∫_ (-∞)^y^(-(x - μ)^2/(2 σ^2))/((2 π)^(1/2) σ) x ; Underscript[lim, y∞] F](images/assuming_7.gif "F = ∫_ (-∞)^y^(-(x - μ)^2/(2 σ^2))/((2 π)^(1/2) σ) x ; Underscript[lim, y∞] F")
![Underscript[lim, y∞] 1/((2 π)^(1/2) σ) If[Re(σ^2) >0, & ... 3309;^(-(x - μ)^2/(2 σ^2)), {x, -∞, y}, AssumptionsRe(σ^2) ≤0]]](images/assuming_8.gif "Underscript[lim, y∞] 1/((2 π)^(1/2) σ) If[Re(σ^2) >0, & ... 3309;^(-(x - μ)^2/(2 σ^2)), {x, -∞, y}, AssumptionsRe(σ^2) ≤0]]")
With the assumption that sigma=0
![Assuming[σ>0, F = ∫_ (-∞)^y^(-(x - μ)^2/(2 σ^2))/((2 π)^(1/2) σ) x ; Underscript[lim, y∞] F]](images/assuming_9.gif "Assuming[σ>0, F = ∫_ (-∞)^y^(-(x - μ)^2/(2 σ^2))/((2 π)^(1/2) σ) x ; Underscript[lim, y∞] F]")
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