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subhadra
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expansion of e^z/z*(1-z)
How do I find the Laurent series expansion of e^z/z*(1-z)?
- Thread starter subhadra
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In summary, a Laurent series expansion is a mathematical representation of a complex function that includes both positive and negative powers of the complex variable. This makes it more versatile than a Taylor series expansion, which only includes positive powers. It is calculated by finding the coefficients of the positive and negative powers using methods such as the Cauchy integral formula or the residue theorem. The principal part of a Laurent series refers to the terms with negative powers and provides important information about the behavior of the function near singularities. It is useful for functions with poles or singularities and in complex analysis and engineering applications.
expansion of e^z/z*(1-z)
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tiny-tim
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hi subhadra! 
(try using the X2 button just above the Reply box)
show us what you've tried, and where you're stuck, and then we'll know how to help!
(try using the X2 button just above the Reply box
)show us what you've tried, and where you're stuck, and then we'll know how to help!
FAQ: How do I find the Laurent series expansion of e^z/z*(1-z)?
1. What is a Laurent series expansion?
A Laurent series expansion is a mathematical representation of a complex function in terms of powers of the complex variable, including both positive and negative powers. This allows for the representation of functions with poles or singularities, which cannot be represented by traditional power series expansions.
2. How is a Laurent series expansion different from a Taylor series expansion?
A Taylor series expansion only includes positive powers of the complex variable, while a Laurent series expansion includes both positive and negative powers. This makes a Laurent series more versatile and able to represent a wider range of functions.
3. How is a Laurent series expansion calculated?
A Laurent series expansion is calculated by finding the coefficients of the positive and negative powers of the complex variable in the function's power series representation. This can be done using various methods such as the Cauchy integral formula or the residue theorem.
4. What is the significance of the principal part in a Laurent series expansion?
The principal part in a Laurent series expansion refers to the terms with negative powers of the complex variable. These terms represent the singularities of the function and can provide important information about the behavior of the function near these points.
5. When is a Laurent series expansion useful?
A Laurent series expansion is useful in situations where a function has poles or singularities, as it allows for a more accurate representation of the function near these points. It is also useful in complex analysis and engineering applications for solving differential equations and approximating functions.