Inverting Series: Solving for z in Powers of x

[qbslug]

How do you invert a series? Can't find any information on it.
I need to invert a function and solve for z in powers of x

$$x = \sum_{n=1}^\infty\\b_n z^n$$

I'm guessing we can just say

$$z = \sum_{n=1}^\infty\\a_n x^n$$

But if this is right I don't know why its right or if it can be proven.
I would like to know the general idea behind inverting series functions and why it works.

 

[jackmell]

Here's a link:

http://mathworld.wolfram.com/SeriesReversion.html

 

[HallsofIvy]

Pretty darn complicated! Note that the term "reversion" is used since "inversion" could be confused with finding the reciprocal, 1/z.

 

[qbslug]

Ok thanks I guess I wasn't getting much information about it because I was referring to it as "inversion"

 

[CellCoree]

Inverting a series involves finding the inverse function of a power series, where the roles of the independent and dependent variables are switched. In your example, you are trying to solve for z in terms of x, which means finding the inverse function of the series with z as the independent variable.

To invert a series, we use the process of finding the compositional inverse of a function. This involves manipulating the series algebraically to isolate the variable we are solving for (in this case, z) on one side of the equation. This can be done by using algebraic techniques such as factoring, expanding, and rearranging terms.

Once we have the series in the form z = f(x), we can then use the Taylor series expansion to express f(x) as a power series, giving us the inverted series of x in terms of z.

The reason this works is because a power series is an infinite polynomial, and by finding the compositional inverse, we are essentially reversing the process of substitution and expanding the polynomial in terms of the new variable.

It is important to note that not all series can be inverted, and the process can be quite complex for higher degree polynomials. It is also important to check the convergence of the inverted series to ensure that it is valid.

In summary, inverting a series involves finding the inverse function of a power series by manipulating the series algebraically and using the Taylor series expansion. It is a complex process that requires a good understanding of algebra and series convergence.