dwsmith said:$$
\frac{1}{z^2(1-z)} = \frac{A}{z^2}+\frac{B}{1-z}
$$I can't figure out how to decompose this fraction.
Complex partial fractions are a mathematical concept used to break down a complex rational function into simpler fractions. This is done by expressing the function as a sum of simpler rational functions with denominators of lower degrees.
Complex partial fractions are important because they allow for easier integration and simplification of complex functions in calculus. They can also be used to solve differential equations and perform other mathematical operations.
To find the partial fraction decomposition of a complex function, you must first factor the denominator of the function into irreducible quadratic factors. Then, using algebraic manipulation, you can find coefficients for each term in the decomposition.
A simple partial fraction has a linear denominator, meaning it can be expressed as a single fraction. A complex partial fraction has a quadratic or higher degree denominator, requiring multiple fractions to represent the function.
Complex partial fractions are used in engineering, physics, and other scientific fields to model and analyze complex systems. They are also used in signal processing and control theory to represent and manipulate signals and systems.