I'm trying to understand how Cases work in Mathematica, and I'm a bit curios why i't wont pick up the Sqrt function?
Consider the following expressions
Cases[Erfc[b], _Erfc, Infinity] Cases[a*Erfc[b], _Erfc, Infinity] Cases[a*Sqrt[b], _Sqrt, Infinity] Cases[a*Sqrt[b], _Sqrt, Infinity] Cases[a*Sqrt[a]*Erfc[b], _Erfc, Infinity] Cases[a*Sqrt[a]*Erfc[b], _Sqrt, Infinity] they give respectively
{} {Erfc[b]} {} {} {Erfc[b]} {} Why does Cases fail in the cases that give {}?
In the first case:
Cases[Erfc[b], _Erfc, Infinity] "Infinity" means: {1,Infinity}. But Erfc[b] is at level 0. Therefore, you need:
Cases[Erfc[b], _Erfc, {0,Infinity}] Further, the next problem:
Cases[a*Sqrt[b], _Sqrt, Infinity] The full form of Sqrt[b] is Power[..] (You may use "FullForm" to see the full form). Therefore, you need:
Cases[a*Sqrt[b], _Power, Infinity] To get:
{Sqrt[b]} Then the case of:
Cases[a*Sqrt[a]*Erfc[b], _Sqrt, Infinity]
Because arguments are evaluated before sending them to functions, a*Sqrt[a] is simplified to a^(3/2) and by:
Cases[a*Sqrt[a]*Erfc[b], _Power, Infinity] you get:
{a^(3/2)} Check FullForm, arg of Cases is evaluated before matching:
FullForm[a*Sqrt[b]] Cases[HoldForm[a*Sqrt[b]], _Sqrt, Infinity] MatchQ[f[x], f[x]] Symbol === Head[f] Cases[f[x], _f, Infinity] Cases[{f[x]}, _f, Infinity] 
Cases[Erfc[b], _Erfc, Infinity] gives {}? $\endgroup$ Head being a symbol, see the edit, not sure why $\endgroup$ Cases[Erfc[b], _Erfc, {0, Infinity}]. The level spec Infinity is equivalent to {1, Infinity}. $\endgroup$ Try HoldForm
Cases[HoldForm[ Erfc[b]], _Erfc, Infinity] Cases[HoldForm[ a*Erfc[b]], _Erfc, Infinity] Cases[HoldForm[ a*Sqrt[b]], _Sqrt, Infinity] Cases[HoldForm[ a*Sqrt[b]], _Sqrt, Infinity] Cases[HoldForm[ a*Sqrt[a]*Erfc[b]], _Erfc, Infinity] Cases[HoldForm[ a*Sqrt[a]*Erfc[b]], _Sqrt, Infinity] 
FullForm[Sqrt[x]]. It returnsPower[x, Rational[1, 2]]which corresponds tox^(1/2), and either you have to prevent evaluation or match this form. $\endgroup$