Expanding 1-2x^{-1} (1-x^2a^{-1}) in Ascending Powers of x
In this article, we will explore the methods to expand the expression (1 - 2x^{-1}(1 - x^2a^{-1})) in ascending powers of (x) for different cases of the parameter (a). We will utilize partial fractions and infinite series to derive the expansions.
Overview of the Problem
The given expression is:
[ y 1 - 2x^{-1} cdot (1 - x^2a^{-1}) frac{1}{1 - 2x(1 - x^2a^{-1})} frac{1}{1 - 2xa - x^2} ]
Case 1: When (a 0)
In this scenario, the expression simplifies significantly. Here's the detailed expansion:
Given: ( y_{a -1} frac{1}{1 - 2x cdot -x^2} frac{1}{2x^3left(1 - frac{1}{2x}right)} )
This can be further resolved as:
[ y_{a -1} frac{1}{2x^3} left(1 frac{1}{2x} frac{1}{4x^2} cdots right) ]
Expanding the series, we get:
[ y_{a -1} frac{1}{2x^3} frac{1}{4x^4} frac{1}{8x^5} cdots sum_{n0}^{infty} frac{1}{2^{n 1} x^{n 3}} ]
Case 2: When (a eq 0)
Here, we introduce (b sqrt{a}) for the sake of convenience. The expression becomes:
[ y_{a eq 0} frac{1}{1 - 2x cdot b^2 - x^2} ]
Applying partial fractions, we obtain three terms:
[ y_{a eq 0} frac{4}{4b^2 - 1} cdot frac{1}{1 - 2x} - frac{1}{2b cdot (2b - 1) cdot (b - x)} - frac{1}{2b cdot (2b 1) cdot (b x)} ]
These terms can be expanded using the geometric series formula:
[ y_{a eq 0} sum_{n0}^{infty} left[ frac{2^{n-2}}{4b^2 - 1} - frac{(-1)^n}{2 cdot (2b - 1) cdot b^{n-2}} - frac{1}{2 cdot (2b 1) cdot b^{n-2}} right] cdot x^n ]
Case 3: When (a
Setting (b sqrt{-a}) for this case, we get:
[ y_{a
The expression involves two terms:
[ y_{a
Expanding these terms with the infinite series, we write:
[ y_{a
This results in the following simplified form:
[ y_{a
Conclusion
In this article, we have provided different series expansions for the given expression based on the value of the parameter (a). These expansions help in understanding the behavior of the expression in various scenarios and can be useful in various mathematical and engineering applications.
Additional Resources
For further exploration, you may refer to the following resources:
Partial Fractions in the Complex Plane Geometric Series