Basic Data Analysis for Time Series with R

7 REVIEW OF TRANSCENDENTAL FUNCTIONS AND COMPLEX NUMBERS

7.1 BACKGROUND

A significant part of this book involves the fitting of periodic functions to periodic data. Many derivations in later chapters will also require understanding of the relationships between periodic functions and exponential functions. These are very powerful techniques, but they are based on some deep mathematics. Although the focus here is not on the proofs behind these methods, even the use of these methods, if any insight is desired, requires some very advanced algebraic and trigonometric manipulations. Besides, this is very cool mathematics.

One key formula associated with many of the manipulations performed throughout the rest of the book is due to Euler: e^ix = cos (x) + i · sin(x). A standard rationale for this formula, based on power series and complex numbers, will be given. Much of the mathematics in later chapters involves moving back and forth between the exponential and trigonometric forms of certain expressions. It is worth noting that a special case of this formula, e^iπ + 1 = 0, is considered by many to be the most beautiful formula in mathematics.

7.2 COMPLEX ARITHMETIC

7.2.1 The Number i

Many students who take statistics may have never been exposed to complex numbers at all. Complex numbers can be loosely considered linear combinations of real and imaginary numbers. The core of imaginary numbers is the quantity . Any number of the form a + bi, where a and b are real, is a complex number. ...