EULER'S IDENTITY

This important identity, named for the Swiss mathematician Leonhard Euler (pronounced Oiler) (1707-1783), can be proved using the power series expansion of cos ?, sin ?, and e ^{j ?},

Since j ² = ?1, the last identity can be written as

Finally,

that is, Euler's identity,

The above identity, in turn, allows us to express the sine and cosine functions in terms of exponentials, as follows. Consider the function obtained by replacing j with ? j in Equation (A.4),

Adding Equations (A.4) and (A.5) pairwise and dividing through by 2,

Likewise, subtracting Equation (A.5) and (A.4) pairwise and dividing through by j2,

We use these expressions in various parts of the book.

THE UNDERDAMPED RESPONSE

The expression for the underdamped response is obtained by substituting Equations (9.20) into Equation (9.10),

Factoring out the common term e ^{?? t},

We wish to prove that this expression can be put in the form

The proof is based on Euler's identity of Equation (A.4), which in the present case becomes

Substituting into Equation (A.9) and collecting yields, on the one hand,

Expanding Equation (A.10) yields, on the other hand,

For the above expressions to be equivalent, the coefficients of the terms cos ? _d t and sin ? _d t must be equal,

Squaring each side of Equations (A.12a, b) and adding terms pairwise,

Using the identities cos ² ? + sin ² ? =...