Laurent Series

Definition

A Laurent series is an infinite series expansion of a complex-valued function that is valid in an annular region around a point, allowing for both positive and negative powers of the variable. It is a generalization of the Taylor series, which is valid only in a disk around the point.

5 Must Know Facts For Your Next Test

1. The Laurent series expansion of a function $f(z)$ around a point $z_0$ is given by $f(z) = \sum_{n=-\infty}^\infty a_n(z-z_0)^n$, where the coefficients $a_n$ are determined by the derivatives of $f(z)$ at $z_0$.
2. The Laurent series is valid in an annular region around $z_0$ that excludes any singularities of $f(z)$, unlike the Taylor series which is only valid in a disk around $z_0$.
3. The Laurent series can be used to study the behavior of a function near a pole or essential singularity, where the Taylor series may not converge.
4. The coefficients $a_n$ in the Laurent series can be calculated using the Cauchy integral formula, which relates the coefficients to the values of the function on a contour surrounding the point $z_0$.
5. The Laurent series is an important tool in complex analysis, as it allows for the classification of singularities and the study of the behavior of functions near these singularities.

Review Questions

• Explain how the Laurent series is a generalization of the Taylor series, and describe the key differences between the two.
  • The Laurent series is a generalization of the Taylor series, as it allows for both positive and negative powers of the variable, whereas the Taylor series only includes positive powers. This means that the Laurent series is valid in an annular region around a point, rather than just a disk around the point like the Taylor series. The Laurent series can be used to study the behavior of a function near a pole or essential singularity, where the Taylor series may not converge. Additionally, the coefficients of the Laurent series are determined using the Cauchy integral formula, rather than the derivatives of the function at the point of expansion like the Taylor series.
• Describe the process of determining the coefficients of a Laurent series expansion and explain how this differs from the process for a Taylor series.
  • The coefficients $a_n$ in the Laurent series expansion $f(z) = \sum_{n=-\infty}^\infty a_n(z-z_0)^n$ are calculated using the Cauchy integral formula, which relates the coefficients to the values of the function on a contour surrounding the point $z_0$. Specifically, $a_n = \frac{1}{2\pi i} \oint_{\gamma} \frac{f(z)}{(z-z_0)^{n+1}} dz$, where $\gamma$ is a closed contour around $z_0$. This is in contrast to the Taylor series, where the coefficients are determined directly from the derivatives of the function at the point of expansion, $a_n = \frac{f^{(n)}(z_0)}{n!}$. The use of the Cauchy integral formula allows the Laurent series to be valid in an annular region around $z_0$, rather than just a disk like the Taylor series.
• Explain how the Laurent series can be used to classify the singularities of a function and describe the relationship between the coefficients of the Laurent series and the type of singularity.
  • The Laurent series is an important tool in the classification of singularities of a function. By examining the coefficients of the Laurent series expansion of a function around a point, one can determine the type of singularity present. If the series contains only positive powers of $(z-z_0)$, then the function has a removable singularity at $z_0$. If the series contains a finite number of negative powers of $(z-z_0)$, then the function has a pole of order equal to the degree of the lowest negative power. If the series contains an infinite number of negative powers, then the function has an essential singularity at $z_0$. The specific values of the coefficients, particularly the coefficients of the negative powers, provide further information about the nature of the singularity. This classification of singularities using the Laurent series expansion is a crucial part of complex analysis.

Related terms

Power Series: A power series is an infinite series of the form $\sum_{n=0}^\infty a_n(x-a)^n$, where $a_n$ are the coefficients and $a$ is the center of the series.

Taylor Series: A Taylor series is a power series expansion of a function about a point, where the coefficients are derived from the derivatives of the function at that point.

Analytic Function: An analytic function is a function that can be expressed as a convergent power series in a neighborhood of every point in its domain.