What are some guidelines on which function to select from among the superficially similar functions Defer, Hold, Unevaluated, and Inactivate?
Consider, for example:
Defer[Defer[Sum[1/k^2, {k, 1, Infinity}]]] Evaluate @@ % Hold[Sum[1/k^2, {k, 1, Infinity}]] ReleaseHold[%] Unevaluated[Sum[1/k^2, {k, 1, Infinity}]] Evaluate @@ % Inactivate[Sum[1/k^2, {k, 1, Infinity}], Sum] Activate[%] The SO duplicate Preventing evaluation of Mathematica expressions does a nice job of explaining the differences among and use-cases for Defer, Hold, and Unevaluated.
Here, I can give at least one use-case for Inactivate for which the other functions are less useful (if not completely useless). There are plenty of situations in which I want to control the algebraic steps that Mathematica does automatically: sometimes I'm doing derivations---using Mathematica to get all of the negative signs right on the first try---where I want to see all of the steps in order that I can write them nicely in a paper later; sometimes I want to show the steps to some students; etc.
The problem is, of course, that Mathematica has all kinds of built-in automatic simplifications, and Simplify does what it wants. So I use Inactivate on some of the functions/Heads in the expression in order to stop the unwanted automatic simplifications while still being able to use the rest of the automatic simplifications.
Here's a contrived example, taken from a previous answer of mine. We're trying to show that the expression
(1 - Tan[x])/(Sin[x] - Cos[x]) simplifies to
- 1/Cos[x] or
- Sec[x] but Sin[x]/Cos[x] automatically simplifies to Tan[x], so we can't do a Replacement to show the step:
(1 - Tan[x])/(Sin[x] - Cos[x]) /. Tan[x] :> Sin[x]/Cos[x] (* (1 - Tan[x])/(-Cos[x] + Sin[x]) *) Instead, we can Inactivate all of the trigonometric functions:
expr = Inactivate[(1 - Tan[x])/(Sin[x] - Cos[x]) /. Tan[x] :> Sin[x]/Cos[x], Tan | Sin | Cos] expr // Simplify % // Activate (* (1 - Inactive[Sin][x]/Inactive[Cos][x])/(-Inactive[Cos][x] + Inactive[Sin][x]) *) (* -(1/Inactive[Cos][x]) *) (* -Sec[x] *) Of course, we could have Replaced Heads instead, using sin, cos, etc., but Activate/Inactivate is a nice programmatic way of doing that that doesn't require a whole bunch of replacement rules, and you have the power of Activateing in steps:
Activate[1 - Inactive[Sin][x]/Inactive[Cos][x], Cos] (* 1 - Sec[x] Inactive[Sin][x] *) which can be quite nice.
HoldForm[]vs.Defer[], the output of the latter can be copied into a new cell and evaluated, while the former's output will still remain held after copying into a new cell. $\endgroup$Inactivate). $\endgroup$Inactivatefit? $\endgroup$