Series and Integrals

When you press the SIN or LOG key on your scientific calculator, it almost instantly returns a numerical value. What is really happening is that the microprocessor inside the calculator is summing a series representation of that function, similar to the series we have already encountered for sinx, cosx, and in Chapters 3 and 4. Power series (sums of powers of x) and other series of functions are very important mathematical tools, both for computation and for deriving theoretical results. 

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Hanna, R. Fourier Series and Integrals of Boundary Value Frohlems, Wiley, New York (1982). 

Carslaw, H. S., Introduction to the Theory of Fourier s Series and Integrals (third edition), Dover Publications, New York (1952). 

DyM] Dym, H. and H. McKean, Fourier Series and Integrals (Probability and Matbematical Statistics, Vol. 14) Academic Press, San Diego, 1972. 

In one of the first articles on this subject [8], the general analytical solution of Eq. (3) was derived. This general solution is easy to find, but it contains infinite series and (integration) constants that depend on the boundary conditions. Those were determined for the central cells of square and triangular arrays, using the boundary collocation method [8]. More recent publications on this subject are based mostly on complete numerical solution using finite-element methods. 

Bataillard et al. (1993) described the integration of a series of thermocouples (thermopiles) on a silicon chip in order to increase the voltage output. The chip consists of an array of p-type silicon/aluminium thermocouples, connected in series and integrated in a n-type silicon epoxy layer grown on the silicon wafer. The overall size of the chip is only 5x5 mm. Several enzymes are immobilized on the array, and the chips are inserted into a FIA system. Glucose is monitored with coimmobilized glucose oxidase and catalase ranging from 2 to 100 mmol/1, urea (1-1000 mmol/1) with immobilized urease and penicillin (1-1000 mmol/1) with immobilized //-lactamase. 

Expanding exp (jijr kT) of exp [—U(r)/kT] into a power series and integrating term by term, he obtained... 

David L. Powers, Boundary Value Problems, Harcourt/Academic Press, New York, 1999. This book includes a 40-page chapter on Fourier series and integrals. 

Spreading (constriction) resistance is an important thermal parameter that depends on several factors such as (1) geometry (singly or doubly connected areas, shape, aspect ratio), (2) domain (half-space, flux tube), (3) boundary condition (Dirchlet, Neumann, Robin), and (4) time (steady-state, transient). The results are presented in the form of infinite series and integrals that can be computed quickly and accurately by means of computer algebra systems. Accurate correlation equations are also provided. 

Chapter 4 and Chapter 5 introduce Sturm-Liouville problems and Fourier series and integrals, respectively. These topics contain essential background... 

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