📂Lemmas

What is a Closed Form in Mathematics?

Glossary ¹

In mathematics, a closed form expression of a function refers to an expression constructed from a finite number of symbols and arithmetic operations $+, -, \cdot, \div$, as well as several well-known functions. These well-known functions include radicals $\sqrt[n]{\cdot}$, exponential functions $\exp$, logarithmic functions $\log$, trigonometric functions $\sin , \cos$, factorials $\cdot !$, etc. Typically, sums $\sum$, products $\prod$, integrals $\int$, and limits $\lim$ are not included.

Examples

The sixth approach in the references seems to offer a more mathematically rigorous definition, but the necessity to know to such an extent is debatable.

When it Exists

Taking an infinite geometric series as an example $|r|<1$, it has a closed form in the following case: $$ \sum_{n=1}^{\infty} a r^{n-1} = { a \over {1-r}} $$ Here, specifically, the right-hand side is in closed form, and the presence of expressions not in closed form, like the left-hand side, does not affect the existence of a closed form. Therefore, when we say “a closed form does not exist,” it usually means “a closed form has not been found yet,” and it’s less common to mean “it has been proven that a closed form does not exist.”

Certainly, proving non-existence is much harder than proving existence, and the fact that “equations of the fifth degree or higher do not have a solution formula” is well-known. Regardless of the degree, the existence of a solution is guaranteed by the Fundamental Theorem of Algebra, but it cannot be expressed in closed form.

When it Does Not Exist

This is the case for most differential equations besides some textbook examples. For three bodies with mass $m_{k}$ and the positions denoted as $r_{k}$, under the influence of gravitational acceleration $G$, solving for the solutions of this coupled dynamic system is known as the three-body problem. $$ \begin{align*} \ddot{r_{1}} =& G \left( m_{2} {{ r_{2} - r_{1} } \over { \left| r_{2} - r_{1} \right|^{3} }} + m_{3} {{ r_{3} - r_{1} } \over { \left| r_{3} - r_{1} \right|^{3} }} \right) \\ \ddot{r_{2}} =& G \left( m_{3} {{ r_{3} - r_{2} } \over { \left| r_{2} - r_{2} \right|^{3} }} + m_{1} {{ r_{1} - r_{2} } \over { \left| r_{1} - r_{2} \right|^{3} }} \right) \\ \ddot{r_{3}} =& G \left( m_{1} {{ r_{1} - r_{3} } \over { \left| r_{1} - r_{3} \right|^{3} }} + m_{2} {{ r_{2} - r_{3} } \over { \left| r_{2} - r_{3} \right|^{3} }} \right) \end{align*} $$ In the 1890s, Poincaré thought it would be impossible to solve this unless some significant new mathematics was introduced, but contrary to his expectation, within less than 20 years, the young Finnish mathematical astronomer Karl Sundman managed to solve the problem in the form of an infinite series using the mathematical techniques available at the time. ² The issue is that the convergence rate of this series is so slow that to calculate a movement of a reasonable time span, it requires terms proportional to $10^{8000000}$, which is clearly impractical.

From such examples, we can see why closed forms are useful, as the existence of a closed form implies a significantly reduced time complexity in calculating the function values algorithmically. To illustrate, using the famous anecdote of Gauss as a child, faced with the problem of finding the sum of an arithmetic sequence $$ f(n) = \sum_{k=1}^{n} k $$ where most people would approach with a repetitive algorithm from $1 + 2 + \cdots + n$ to $O(n)$, Gauss solved it with the cost of $O(1)$ by using its closed form $$ f(n) = {{ n(n+1) } \over { 2 }} $$