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A closed-form expression is a mathematical process that can be completed in a finite number of operations. Closed-form expressions are of interest when trying to develop general solutions to problems.
A closed-form solution is a general solution to a problem in the form of a closed-form expression. A closed-form solution is nearly always desirable because it means that a solution can be found efficiently. Unfortunately, closed-form solutions are not always possible. Many mathematicians concern themselves with finding closed-form solutions to open problems, and in lieu of that, proving whether or not a closed-form solution is possible.
Of course, an operation can be defined to "cheat" the requirements of a closed-form expression. For example, define operation to solve any problem in existence. Voila! Now every single problem in existence has a closed-form solution! Of course, this would be absurd.
It is for this reason that closed-form expressions and solutions are typically restricted to having the following operations:
Addition, subtraction, multiplication, and division
Exponents and logarithms
Trigonometric functions and inverse trigonometric functions
Limits and other operations that involve an infinite number of terms or operations are not permitted. Finite Sums and Products (using the and symbols, respectively) are also generally avoided.