Advanced Calculus

About the Book

The book begins with a discussion of finding the equation of a curve that passes through a fixed number of points. This technique is called curve fitting. This concept is then generalized to finding the equation of a curve that passes through an infinite number of points. This concept is in turn generalized to finding the equation of a curve that passes through a given curve over a specific interval. Such an equation and its curve that can be found to pass through the given curve over a particular interval is called a Fourier series. A Fourier series consists of an infinite number of sine and cosine terms added together in an infinite series whose coefficients must be determined. Such an infinite series of terms can be made to approximate the curve of an elementary function over a particular interval arbitrarily close. Also, since the terms of the Fourier series are Trigonometric terms the curve of their sum namely the Fourier series is periodic with a definite period. This brings us to the concepts of the periodic function. The curve of such a function keeps repeating itself over the same intervals. Numerous examples are provided for finding the Fourier series of various elementary functions over given intervals. The following concepts are also discussed-functions in general form, composite functions, and odd and even functions. The Fourier series of each of these types of functions is also found. Next. The equations for the Fourier series and its coefients are generalized to the complex number system. This allows is to derive the Fourier transform. Everything is logically derived with all of the steps included.

About the Author

Edward Walsh graduated from Moorhead State College in Moorhead, Minnesota with a B.A. Degree in mathematics. He took a year of applied mathematics including coarses in vector and tensor analysis; partial differential equations; Fourier analysis; and complex analysis.