Many functions cannot be expressed as a power series a_{0} + a_{1}x + a_{2}x^{2} + …. For example, any functions that tend to infinity as x → 0 have value a_{0} when x →0, so are not expressible as power series. Functions with **fractional powers** of x, like x^{½}, are also problematic, because of **branching behavior** at zero (i.e., they are multivalued at zero and so are not “true” functions at that point). In fact, most algebraic functions have some kind of multi-valued points and so cannot fit a traditional power series, where one value must be placed for every *x* in the series. The solution is a **fractional power series**, the discovery of which is credited to Isaac Newton. Puiseux (1850) applied more rigor to the topic; it is for this reason that fractional power-series expansions of algebraic functions are called *Puiseux expansions* [1], Puiseaux series [2], or sometimes, the *Newton-Puiseaux method*.

The method has many **practical applications**, especially in engineering. For example, it can be used in thermodynamics to solve systems of fractional differential equations [3].

## Formula for the Fractional Power Series

The **general formula** for the fractional power series is

Where r_{k} are rational numbers (rational numbers can be expressed as fractions).

The series can be rewritten as a finite sum of a traditional power series. The powers of x get rewritten as multipliers:

The result is that the behavior of y, in the neighborhood of x = 0, acts like a finite sum of fractional powers.

## Example

If y^{2}(1 + x)^{2}, then:

Near x = 0, y will behave similarly to x^{½}.

## References

[1] Puiseux Series and Algebraic Solutions of First Order Autonomous AODEs — A MAPLE Package.

[2] Stillwell, J. (2010). Mathematics and Its History. 3rd Edition. Springer.

[3] Ren, F. & Hu, Y. (2018). The fractional power series method an efficient candidate for solving fractional systems. Thermal Science 22(4):1745-1751.