Math Problem Statement
evaluate 1/(z-1)(z-2) is valid for the region mod(z-1)<1 using laurents series
Solution
To evaluate the function using a Laurent series that is valid in the region , follow these steps:
Step 1: Express the Function as a Sum of Partial Fractions
The given function can be decomposed into partial fractions: To find the constants and , multiply both sides by and set the equation equal to the numerator: Expanding and equating coefficients: From this, we equate the coefficients of and the constant term:
Step 2: Solve for and
From , we have .
Substitute into the second equation: Then, .
So, the partial fraction decomposition is:
Step 3: Expand Each Term in the Region
We can now expand each term in a Laurent series. Since , we consider the series expansion around :
For the first term: This is already in a simple form and is the first term of the Laurent series.
For the second term , rewrite it by factoring out : Expand this using the geometric series for : This is valid for .
Step 4: Combine the Series
Now combine the two series: This is the Laurent series expansion of the function valid in the region .
Final Answer
Would you like more details or have any questions?
- How would the Laurent series change for a different region, such as ?
- What are the general steps to perform partial fraction decomposition?
- How can the geometric series expansion be generalized for other functions?
- What are the conditions for the convergence of the Laurent series in this region?
- How would the series expansion look if the region were ?
Tip: When working with Laurent series, always carefully consider the region of convergence to determine the correct expansion.
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Math Problem Analysis
Mathematical Concepts
Complex Analysis
Laurent Series
Partial Fraction Decomposition
Formulas
Partial fraction decomposition formula
Geometric series expansion formula
Theorems
Laurent series theorem
Suitable Grade Level
Advanced Undergraduate