Evaluating Limits Using Laurent Series of Zeta and Gamma Functions

The Riemann zeta function and the Gamma function play a crucial role in advanced mathematical analysis. These functions have specific properties that enable us to evaluate various limits, which is essential in both theoretical and applied mathematics. In this article, we will delve into the evaluation of the limit of the ratio of the Riemann zeta function and the Gamma function at specific points using the Laurent series expansions of these functions. This technique is not only mathematically elegant but also widely applicable in various fields, including physics, engineering, and number theory.

Understanding the Laurent Series

The Laurent series is a powerful tool in complex analysis that allows us to represent functions in a neighborhood of a complex number, including points where the function is not analytic. The Laurent series of a function ( f(s) ) around a point ( s s_0 ) is given by:

( f(s) sum_{n-infty}^{infty} a_n (s - s_0)^n )

The series contains negative powers of ( (s - s_0) ), which captures the behavior of the function near singularities.

Evaluation of the Riemann Zeta Function at ( s 1 )

The Riemann zeta function, denoted by ( zeta(s) ), has a simple pole at ( s 1 ) with a residue of 1. A simple pole means that the function behaves like ( frac{1}{s - 1} ) as ( s ) approaches 1. We can rewrite the Laurent series of ( zeta(s) ) at ( s 1 ) as follows:

( zeta(s - 1) -frac{1}{s - 1} text{analytic part} )
Tag: zeta1-s -frac{1}{s} text{analytic part}

This expression reveals the principal term of the Laurent series, which is negatively proportional to ( frac{1}{s} ) as ( s ) approaches 1.

Evaluation of the Gamma Function at ( s 0 )

The Gamma function, denoted by ( Gamma(s) ), also has a simple pole at ( s 0 ) with a residue of 1. The behavior of the Gamma function near ( s 0 ) can be described by the following Laurent series:

( Gammaleft(frac{s}{2}right) frac{2}{s} text{analytic part} )
Tag: GammaBigfrac{s}{2}Big frac{2}{s} text{analytic part}

Once again, this expression captures the principal term of the Laurent series, which is positively proportional to ( frac{2}{s} ) as ( s ) approaches 0.

Evaluating the Limit: ( lim_{s to 0} frac{zeta(s - 1)}{Gammaleft(frac{s}{2}right)} )

With the Laurent series expansions established, we can now evaluate the limit of the ratio of the Riemann zeta function and the Gamma function at ( s 0 ). The steps are as follows:

( lim_{s to 0} frac{zeta(s - 1)}{Gammaleft(frac{s}{2}right)} lim_{s to 0} frac{-frac{1}{s} text{analytic part}}{frac{2}{s} text{analytic part}} )

As ( s ) approaches 0, the analytic parts become negligible, and the expression simplifies to:

( lim_{s to 0} frac{-frac{1}{s}}{frac{2}{s}} lim_{s to 0} frac{-1}{2} -frac{1}{2} )
Tag: lim_{s to 0} frac{-frac{1}{s} Os^2}{2 Os^2} boxed{-1/2}

This elegant result demonstrates the power of using Laurent series to evaluate limits involving special functions.

Conclusion

The evaluation of limits using the Laurent series of the Riemann zeta function and the Gamma function is a critical technique in complex analysis. This method provides a rigorous and systematic approach to determine the behavior of these functions near singularities. Understanding these concepts is not only fundamental for mathematicians but also valuable for students and researchers in related fields.

Further Reading

For those interested in deeper exploration of this topic, consider reading the following resources:

Laurent Series on MathIsFun Riemann Zeta Function on Wikipedia Gamma Function on Wikipedia