Analysis of laurent and fourier series

Thesis Bachelor of Science in Mathematics University of Rizal System-Morong. 2019

EXECUTIVE SUMMARY This study entitled ANALYSIS OF LAURENT AND FOURIER SERIES was conducted at the University of Rizal System, Morong, Rizal during the academic year 2018 - 2019. The purpose of this study was to present and analyze cases of Laurent and Fourier Series. Laurent Series is a representation of a function as a power series which includes terms of negative degree and Fourier Series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. The alternative solution in solving finite series has been applied in two types of series by cases, Laurent Series has two cases, one with both principal part and analytic part and the other with only principal part on the series of expansion and Fourier Series has two cases also, odd function and even function. The researchers gathered examples on books and internet in solving Laurent series and limited from 3 terms sequence up to 10 term sequence. The existing method and alternative solution have an equal result on the 3 terms sequence of the series regardless if the terms are negative and nonnegative. As the number of terms increases the results obtained in Laurent series, on case 1, using existing method also increases. It is because the analytic part of the series has positive exponent that contradicts the increasing number of negative exponent of principal part while on the case 2 of Laurent series, as the number of terms increases the result obtained using the existing method repeats itself several times, since the series are composed of only principal part with negative exponent as its divisor, and as the value of negative exponent increases the number of derivation occurs, the terms tend to be below 0 which can be approximately 0 and that makes the result remain as it is. Moreover, on the alternative solution, the result shows no clear pattern as how it is, it appears that it may be increasing, decreasing or alternating but not equal after the 3 terms sequence. Fourier series. Two functions have been computed for Fourier series, odd function and even function and limited from 3 terms sequence up to 10 terms sequence. On the case of odd function, the even number terms using existing method appear to be 0. It is because the dividend is negative and raised to n that makes it positive and since the series of odd function of Fourier series are composed of negative and positive terms alternately that makes the even number terms using existing method appears to be 0. On the case of even function Fourier series, as the number of terms increases the result obtained using existing method also increases. Even the dividend of the Fourier expansion of the function is the same as in the odd function that doesn't affect the terms because of the coefficient which is not present on the Fourier expansion of odd function. Furthermore, the results obtained using the alternative solution is the same as in the Laurent series, it has no clear pattern and maybe sometimes be increasing, decreasing or alternating with relation to the number of terms. The researchers formulated a generalization which states that for instance, the alternative solution gives an equal result with the existing method. When the given series has three terms, in cases that involved the Laurent series, the given series is equal regardless of the given domain; and on the cases that involves Fourier series, the given series is equal regardless of the given value of x. The result obtained on the n-terms higher than 3 terms, using the existing method and the alternative solution is not equal, both cases of Laurent and Fourier series.