598
Chapter 21
35.
36.
37.
38.
39.
40.
41.
42.
◆
Complex Numbers
(7 20°)2
(1.55 15°)3
兹 57 52°
兹 22 12°
3 兹 38 73°
3 兹 15 89°
兹135 j204
3
56.3 j28.5
兹
21–4
Complex Numbers in Exponential Form
Euler’s Formula
We have already expressed a complex number in rectangular form
a jb
in polar form
r
and in trigonometric form
r(cos j sin )
Our next (and final) form for a complex number is exponential form, given by Euler’s formula.
Euler’s
Formula
This formula is named after
Leonard Euler (1707–83). We
give it without proof here.
re j r(cos j sin )
232
Here e is the base of natural logarithms (approximately 2.718), discussed in Chapter 20, and is the argument expressed in radians.
◆◆◆
Example 25: Write the complex number 5(cos 180° j sin 180°) in exponential form.
Solution: We first convert the angle to radians: 180° rad. So
5(cos j sin ) 5ej
Setting and r 1 in
Euler’s formula gives
Common
Error
e j cos j sin ◆◆◆
Make sure that is in radians when using Eq. 232.
1 j0
1
The constant e arises out of
natural growth, j is the square
root of 1, and is the ratio
of the circumference of a circle
to its diameter. Thus we get
the astounding result that the
irrational number e raised to the
product of the imaginary unit j
and the irrational number give
the integer 1.
Example 26: Write the complex number 3e j2 in rectangular, trigonometric, and polar
forms.
◆◆◆
Solution: Changing to degrees gives us
2 rad 115°
and from Eq. 232,
r3
Section 21–4
◆
599
Complex Numbers in Exponential Form
From Eq. 224,
a r cos 3 cos 115° 1.27
and from Eq. 225,
b r sin 3 sin 115° 2.72
So, in rectangular form,
3e j2 1.27 j2.72
In polar form,
3e j2 3 115°
In trigonometric form,
3e j2 3(cos 115° j sin 115°)
◆◆◆
Products
We now find products, quotients, powers, and roots of complex numbers in exponential form.
These operations are quite simple in this form, as we merely have to use the laws of exponents.
Thus by Eq. 29:
Products
◆◆◆
r1e j1 · r2e j2 r1r2e j (1 2)
233
Example 27:
(a) 2e j3 5e j4 10e j 7
(b) 2.83e j 2 3.15e j4 8.91e j6
(c) 13.5ej3 2.75e j5 37.1e j 2
◆◆◆
Quotients
By Eq. 30:
Quotients
r1e j1
r1
r2
e j (1 2)
r2e j2
234
◆◆◆
Example 28:
8e j5
(a) 2e j3
4e j2
63.8e j2
(b) 4.66ej3
13.7e j5
5.82ej4
(c) 0.592e j3
9.83e j 7
◆◆◆
Powers and Roots
By Eq. 31:
Powers
and Roots
(re j)n r ne jn
235
600
Chapter 21
◆◆◆
◆
Complex Numbers
Example 29:
(a) (2ej3)4 16e j12
(b) (3.85ej2)3 57.1ej6
e j6
ej6
(c) (0.223ej3)2 20.1ej6
2
0.0497
(0.223)
Exercise 4
◆
◆◆◆
Complex Numbers in Exponential Form
Express each complex number in exponential form.
1. 2 j3
2. 1 j2
3. 3(cos 50° j sin 50°)
4. 12 14°
5. 2.5 6
6. 7 p cos j sin 3
3
7. 5.4 12
8. 5 j4
q
Express in rectangular, polar, and trigonometric forms.
9. 5e j3
10. 7e j5
11. 2.2e j1.5
12. 4e j2
Operations in Exponential Form
Multiply.
13. 9e j2 2e j4
15. 7e j 3e j3
17. 1.7e j5 2.1ej2
14. 8ej 6ej3
16. 6.2ej1.1 5.8ej2.7
18. 4ej7 3ej5
Divide.
19. 18ej6 by 6ej3
21. 55ej9 by 5ej6
23. 21ej2 by 7ej
20. 45ej4 by 9ej2
22. 123ej6 by 105ej2
24. 7.7ej4 by 2.3ej2
Evaluate.
25. (3ej5)2
27. (2ej)3
21–5
26. (4ej2)3
Vector Operations Using Complex Numbers
Vectors Represented by Complex Numbers
One of the major uses of complex numbers is that they can represent vectors and, as we will
soon see, can enable us to manipulate vectors in ways that are easier than we learned when
studying oblique triangles.
Take the complex number 2 j3, for example, which is plotted in Fig. 21–7. If we connect that
point with a line to the origin, we can think of the complex number 2 j3 as representing a vector
R having a horizontal component of 2 units and a vertical component of 3 units. The complex number used to represent a vector can, of course, be expressed in any of the forms of a complex number.