Laurent Series Expansion of a Complex Function

In complex analysis, the Laurent series is a representation of a complex function as a power series that includes both positive and negative powers of the variable. This series is particularly useful for understanding the behavior of functions in the neighborhood of singular points. In this article, we will explore the Laurent series expansion of the function (f(z) frac{1}{z - 1}) and (f(z) frac{1}{z - 2}), and how to combine them to form a more complex function.

Understanding the Laurent Series

The Laurent series for a function (f(z)) around a point (z_0) is a series of the form:

[f(z) sum_{n-infty}^{infty} a_n (z - z_0)^n]

where the coefficients (a_n) are given by the formula:

[a_n frac{1}{2pi i} oint_{C} frac{f(z)}{(z - z_0)^{n 1}} dz]

where the integral is taken over a closed contour (C) around the point (z_0).

Decomposing the Function

Consider the function (f(z) frac{1}{z - 1}). To find its Laurent series, we can rewrite it using the geometric series. Recall that:

[frac{1}{1 - x} sum_{n0}^{infty} x^n quad text{for} quad |x|

Therefore, for (|z|

[frac{1}{z - 1} frac{1}{1 - (1 - z)} sum_{n0}^{infty} (1 - z)^n]

Similarly, for (|z|

[frac{1}{z - 2} frac{1}{z} cdot frac{1}{1 - frac{2}{z}} frac{1}{z} sum_{n0}^{infty} left(frac{2}{z}right)^n quad text{for} quad left|frac{2}{z}right|

This simplifies to:

[frac{1}{z - 2} sum_{n0}^{infty} frac{2^n}{z^{n 1}}]

Combining the Functions

Now, let's consider the combined function:

[g(z) frac{1}{z - 1} - frac{1}{z - 2}]

Substituting the decompositions, we get:

[g(z) sum_{n0}^{infty} (1 - z)^n - sum_{n0}^{infty} frac{2^n}{z^{n 1}}]

It is important to note that these series must converge within their respective radii of convergence:

[|1 - z|

[|z| > 2 quad text{for the second series}]

These conditions ensure that the series converge to the given functions. However, for a more general expansion, we can use a single Laurent series that converges in a larger annulus.

Laurent Series for a Combined Function

To find the Laurent series for the combined function (g(z)), we can use the geometric series expansions:

[g(z) sum_{n0}^{infty} (1 - z)^n - sum_{n0}^{infty} frac{2^n}{z^{n 1}}]

Combining the two series, we have:

[g(z) sum_{n0}^{infty} (1 - z)^n - sum_{n0}^{infty} frac{2^n}{z^{n 1}}]

For the series to converge, we need both conditions to be satisfied. This means that the function (g(z)) will have a Laurent series that converges in the annulus (1

Conclusion

In conclusion, the Laurent series expansion of a complex function can provide valuable insights into the behavior of the function in the neighborhood of its singular points. The specific example of combining two geometric series to form a more complex function demonstrates the utility and importance of the Laurent series in complex analysis. By understanding the convergence conditions and the series representations, we can better analyze and manipulate complex functions in various applications.

Keywords

Laurent Series, Geometric Series, Complex Analysis