Simplify

Simplify is command, operator in the Mathematica language.

Usually, the call of this routine has form

Simplify[expresson]

or

Simplify[expresson, conditions]

Several conditions can be combined, for example

conditions$$=$${condition1, condition2, ..}

in the most of cases, Simplify returns expression, equivalent to its argument; and often, it is written in a form, shorter than its argument.

Example of inconditional simplification

Simplify[2 Sin[x] Cos[x]]

returns

Sin[2 x]

Example of conditional simplification

Sometimes, the simplification is valid only for certain range of values of parameters. The simple example is below.

f[x_] = Sqrt[1 + x] Sqrt[1 - x]

g[x_] = Simplify[f[x]]

The last line returns the same expression as

$$\sqrt{1+x}\sqrt{1-x}$$

Specification of $$x$$ may allow the simplification:

h[x_] = Simplify[f[x], x > 0]

$$\sqrt{1-x^2}$$

that coincides with initial expression for positive $$x$$, and in this sense is correct.

However, the result of the conditional simplification may be not valid for values of parameters out of range, declared at the call of Simplify:

g[-1.+I] gives the same as f[-1.+I], id est,

1.27202 + 0.786151 I

while h[-1.+I] gives

1.27202 - 0.786151 I

(In Mathemaica, capital "I" denotes $$\mathrm i=\sqrt{-1}$$.

Not perfect

In some cases, the rules, used in the implementation of the Simplify command, are not sufficient to perform the simplification.

For example,

Simplify[Integrate[(BesselJ[0, BesselJZero[0,1] p]/(1-p^2) )^2 p , {p,0,Infinity}]]

gives complicated expression

-(1/2) Sqrt[Pi] MeijerG[{{}, {1/2}}, {{0, 1}, {0}}, BesselJZero[0, 1]^2]