Laurent Series

The Laurent series is a foundational concept used for complex analysis in mathematical sciences. Karl Weierstrass first used the series in a research paper in 1841. However, he could not publish his research about the series in his lifetime. What makes the series unique and vital to complex analysis is its negative degree.

The Laurent series is used in complex analysis to deal with the singularities of a complex function. The complementary concept of the Laurent series is the Taylor series. The Laurent series deals with the complex singularities, while the Taylor series explains the analytical part of a specific function. Mathematical sciences consider these two series as the essential tools for completing any complex analysis function.

Convergent Laurent Series

As explained before, the Laurent series consists of terms consisting of both positive powers and negative powers. The Laurent series converges when these two powers converge. We can witness a convergent Laurent series on the open annulus, and we can determine the value of convergence radii using this characteristic of the Laurent series. The Laurent also allows us to determine factors and find values that the Taylor series cannot solve.

Definition of Laurent Polynomial

Laurent polynomials are unique elements of mathematical sciences that are vital for complex analysis. Laurent polynomials are a series of terms with a finite number of non-zero elements, and the polynomial is similar to an algebraic polynomial. However, an algebraic polynomial only consists of positive or null power terms, while Laurent polynomials can consist of both positive and negative powers. So, if a number of positive and negative terms form a combination in a given field, we can call it a Laurent polynomial.

Properties of Laurent Polynomial

• Laurent polynomials are Laurent series consisting of a finite number of non-zero terms. 

• If we invert the X and extend the ring of polynomials (denoted by R[X]), we get the Laurent polynomial ring (denoted by R[X, X^-1]).

• The set of terms in the Laurent polynomial ring is a subset of terms forming the ring of rational functions.

Principal Part in Laurent Series

The parts of a Laurent series expansion consisting of negative powers are the principal part of the series. The principal part is a significant feature of the Laurent series as it makes the series unique. The presence of negative degrees in a complex analysis series is a revolutionary discovery in the world of mathematical sciences.

Multiplication and Sum of Laurent Series

It is impossible to multiply the Laurent series using algebraic or geometric methods, and two convergent Laurent series can consist of annuli that do not overlap. The infinite number of sums forming the expansion of two Laurent polynomials will not necessarily have any terms in common. Two Laurent series can only be multiplied if they consist of a finite number of negative terms, and the multiplication is possible because the algebraic sum is finite in number.

Application of Laurent Series

Not all complex mathematical operations have real-world applications, as most of them are vague concepts used to explain the unknown. However, complex analysis is a practice that is useful in theory and has several applications in the real world. The concept of complex analysis is the foundation of analogue electronic design. The concept also plays a vital role in determining impedance in alternating current circuits that constitute audio amplifiers.