The table below indicates the components that are commonly considered part of various types of mathematical expressions. With some exceptions, an expression can only have components that themselves have components of that type. For example, algebraic expressions can only have constants that are themselves algebraic, and finite sums and products in algebraic expressions can only have algebraic arguments.
| Arithmetic expressions | Polynomial expressions | Algebraic expressions | Closed-form expressions | Analytic expressions | Mathematical expressions | |
|---|---|---|---|---|---|---|
| Constant | Yes | Yes | Yes | Yes | Yes | Yes |
| Elementary arithmetic operation | Yes | Addition, subtraction, and multiplication only | Yes | Yes | Yes | Yes |
| Finite sum | Yes | Yes | Yes | Yes | Yes | Yes |
| Finite product | Yes | Yes | Yes | Yes | Yes | Yes |
| Finite continued fraction | Yes | No | Yes | Yes | Yes | Yes |
| Variable | No | Yes | Yes | Yes | Yes | Yes |
| Integer exponent | No | Yes | Yes | Yes | Yes | Yes |
| Integer nth root | No | No | Yes | Yes | Yes | Yes |
| Rational exponent | No | No | Yes | Yes | Yes | Yes |
| Integer factorial | No | No | Yes | Yes | Yes | Yes |
| Irrational exponent | No | No | No | Yes | Yes | Yes |
| Exponential function | No | No | No | Yes | Yes | Yes |
| Logarithm | No | No | No | Yes | Yes | Yes |
| Trigonometric function | No | No | No | Yes | Yes | Yes |
| Inverse trigonometric function | No | No | No | Yes | Yes | Yes |
| Hyperbolic function | No | No | No | Yes | Yes | Yes |
| Inverse hyperbolic function | No | No | No | Yes | Yes | Yes |
| Root of a polynomial that is not an algebraic solution | No | No | No | No | Yes | Yes |
| Gamma function and factorial of a non-integer | No | No | No | No | Yes | Yes |
| Bessel function | No | No | No | No | Yes | Yes |
| Special function | No | No | No | No | Yes | Yes |
| Infinite sum (series) (including power series) | No | No | No | No | Convergent only | Yes |
| Infinite product | No | No | No | No | Convergent only | Yes |
| Infinite continued fraction | No | No | No | No | Convergent only | Yes |
| Limit | No | No | No | No | No | Yes |
| Derivative | No | No | No | No | No | Yes |
| Integral | No | No | No | No | No | Yes |