A Marriage Of Taylor And Fourier Series To Generate Definite Integrals
DR. TOM OSLER
JAMES ROSADO
MATHEMATICS DEPARTMENT
ROWAN UNIVERSITY
GLASSBORO, NEW JERSEY
JOINT MATHEMATICS MEETING – SAN ANTONIO, TX – 2015
ABSTRACT:
In this presentation, we will demonstrate an unusual method of obtaining closed form solutions to complicated integrals of the form:
Where n is a positive integer.
This method is unusual because the integrals are the result of a process which does not envision the integrals initially; instead, the closed form integral solutions are the result of comparing coefficients of power series and related Fourier Series.
INTRODUCTION
We will utilize the…
To find closed form expressions to definite integrals. Some of these integrals maybe new.
We will demonstrate this methodology by an example.
AN EXAMPLE
Let and substitute . We can express our function g(z) as the sum of a real and imaginary part:
Observe that the line above is a Fourier Series expansion.
Let us make the same substitution for the Taylor
Series expansion of our function g(z) =
This is the Cosine Series Expansion of u(r,θ)
This is the Sine Series Expansion of v(r,θ)
If we equate the results of the pervious two slides, we achieve the sine and cosine series expansions of the real and imaginary parts to g(z).
Notice from the previous slide we have found Fourier Series expansions, but more importantly we have stumbled upon solutions to definite integrals. Recall:
We have achieved our goal of evaluating complicated definite integrals. With some simple manipulation of the three integrals, and keeping ‘r’ fixed and θ is the variable of integration we obtain:
This methodology works best if our original function g(z) can be conveniently decomposed to real and imaginary parts.
Using this method we have obtained solutions to 36 very complicated integrals:
These integrals occur in groups of three, and some maybe new integrals