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In mathematics, an analytical expression (or expression in analytical form) is a mathematical expression constructed using well-known operations that lend themselves readily to calculation. As is true for closed-form expressions, the set of well-known functions allowed can vary according to context but always includes the basic arithmetic operations (addition, subtraction, multiplication, and division), extraction of nth roots, exponentiation, logarithms, and trigonometric functions.
However, the class of expressions considered to be analytical expressions tends to be wider than that for closed-form expressions. In particular, special functions such as the Bessel functions and the gamma function are usually allowed, and often so are infinite series and continued fractions. On the other hand, limits in general, and integrals in particular, are typically excluded.
If an analytic expression involves only the algebraic operations, which are addition, subtraction, multiplication, division and exponentiation with integral or fractional exponents (hence including the extraction of nth roots), then it is more specifically referred to as an algebraic expression.
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