COMPLEX NUMBERS AND EULER’S RELATION - Signals, Systems, and Transforms - page 728

e jy - e -jy

2j

sin y = . (D.13)

The four expressions (D.10) through (D.13) are of such importance that they should
be memorized. In the engineering use of these four expressions, y is usually real;
however, these expressions are valid for y complex.
We will again suppose that s is complex. Then

e s = e a+ jb = e a e jb = e a (cos b + j sin b)

= e a cos b + je a sin b. (D.14)

Thus, the exponential raised to a complex power is itself complex, with the real and
imaginary parts as given in (D.14). For example, for the value of s 1 given previously,

e s 1 = e 2 + j2 = e 2 (cos 2 + j sin 2)

= e 2 cos 114.6° + je 2 sin 114.6° =-3.076 + j6.718,

because 1 rad = 57.30°. This evaluation is performed in M ATLAB ATLAB by

exp (2+2*j)
result: 3.0749 + 6.7188i

Conversion Between Forms

We now consider expressing a complex number in a form other than the rectangular
form, based on the foregoing developments. Euler's relation is given by

e ju

= cos u + j sin u.

Letting A and u be real numbers, we see that

Ae ju = A(cos u + j sin u) = A cos u + jA sin u

= a + jb = s. (D.15)

Thus, a complex number s can be expressed as a real number multiplied by a com-
plex exponential; this form is called the exponential form . For example,

5e jp/6 = 5 cos 30° + j5 sin 30° = 4.33 + j2.50.

This evaluation is performed in MATLAB by
5*exp(j*pi/6)
result: 4.3301 + 2.5i