Why laurent series

The Taylor Series is a good way to begin discussion of the Laurent Series. The Taylor Series is of the form:. This series describes any smooth function as a sum of an infinite power series.

The area this series accurately describes the function is the region of convergence of the Taylor Series. This area is limited by where the series returns a finite answer.

When describing the Laurent Series r is called the radius of convergence. This sum allows a continuous function to be represented by a sum of a discrete set of values. The concepts of region of convergence and translating a discrete signal into a continuous signal relate directly to the Laurent Series. Using a background in the Taylor Series, the Laurent series can be viewed as the complex extension that allows for the existance of poles and can even give a basic description of those poles.

The formula is. The Laurent Series seems to be an analogue to the Taylor Series on the complex plane with the sum going to negative infinity instead of stopping at 0. This is a good way to think of the Laurent Series, as it does basically the same thing. Arfken, G. Orlando, FL: Academic Press, pp.

Korn, G. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, p. Knopp, K. New York: Dover, pp.

Krantz, S. Morse, P. New York: McGraw-Hill, pp. Goodmanson, David and Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.

Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. This would have produced a different Laurent series. We discuss this further in an upcoming example. The following example shows that the Laurent series depends on the region under consideration.

One lesson from this example is that the Laurent series depends on the region as well as the formula for the function. Examples of Laurent Series In general, the integral formulas are not a practical way of computing the Laurent coefficients.