Complex Number (Notes+Solution).pdf

COMPLEX NUMBERS Im 2+3i _4+2i i 0 _i _2-2i

Re

1

3-2i

FIGURE 1

Complex numbers as points in the Argand plane

A complex number can be represented by an expression of the form a
bi, where a and b are real numbers and i is a symbol with the property that i 2
1. The complex number a
bi can also be represented by the ordered pair
a, b and plotted as a point in a plane (called the Argand plane) as in Figure 1. Thus, the complex number i
0
1
i is identified with the point
0, 1. The real part of the complex number a
bi is the real number a and the imaginary part is the real number b. Thus, the real part of 4  3i is 4 and the imaginary part is 3. Two complex numbers a
bi and c
di are equal if a
c and b
d, that is, their real parts are equal and their imaginary parts are equal. In the Argand plane the horizontal axis is called the real axis and the vertical axis is called the imaginary axis. The sum and difference of two complex numbers are defined by adding or subtracting their real parts and their imaginary parts:
a
bi

c
di

a
c

b
di
a
bi 
c
di

a  c

b  di For instance,
1  i

4
7i

1
4

1
7i
5
6i The product of complex numbers is defined so that the usual commutative and distributive laws hold:
a
bi
c
di
a
c
di

bi
c
di
ac
adi
bci
bdi 2 Since i 2
1, this becomes
a
bi
c
di

ac  bd

ad
bci EXAMPLE 1


1
3i
2  5i

1
2  5i
3i
2  5i
2
5i
6i  15
1
13
11i Division of complex numbers is much like rationalizing the denominator of a rational expression. For the complex number z
a
bi, we define its complex conjugate to be z
a  bi. To find the quotient of two complex numbers we multiply numerator and denominator by the complex conjugate of the denominator. EXAMPLE 2 Express the number

1
3i in the form a
bi. 2
5i

SOLUTION We multiply numerator and denominator by the complex conjugate of 2
5i,

namely 2  5i, and we take advantage of the result of Example 1:

Im

1
3i 1
3i 2  5i 13
11i 13 11


2
i 2
2
5i 2
5i 2  5i 2
5 29 29 z=a+bi

i 0

Re

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_i

The geometric interpretation of the complex conjugate is shown in Figure 2: z is the reflection of z in the real axis. We list some of the properties of the complex conjugate in the following box. The proofs follow from the definition and are requested in Exercise 18. Properties of Conjugates

z=a-bi –

z
w
z
w

zw
z w

zn
zn

FIGURE 2 1

2 ■ COMPLEX NUMBERS

 

The modulus, or absolute value, z of a complex number z
a
bi is its distance from the origin. From Figure 3 we see that if z
a
bi, then

Im

@ z=a+bi „b„ „ + „@  œ   „a

bi

= |z|

 z 
sa

b

0

Re

a

2


b2

Notice that zz

a
bi
a  bi
a 2
abi  abi  b 2i 2
a 2
b 2

FIGURE 3

 

zz
z

and so

2

This explains why the division procedure in Example 2 works in general: z w




zw ww




zw

w

2

Since i 2
1, we can think of i as a square root of 1. But notice that we also have
i2
i 2
1 and so i is also a square root of 1. We say that i is the principal square root of 1 and write s1
i. In general, if c is any positive number, we write sc
sc i With this convention, the usual derivation and formula for the roots of the quadratic equation ax 2
bx
c
0 are valid even when b 2  4ac  0: x


b  sb 2  4ac 2a

EXAMPLE 3 Find the roots of the equation x 2
x
1
0. SOLUTION Using the quadratic formula, we have

x


1  s1 2  4
1 1  s3 1  s3 i

2 2 2

We observe that the solutions of the equation in Example 3 are complex conjugates of each other. In general, the solutions of any quadratic equation ax 2
bx
c
0 with real coefficients a, b, and c are always complex conjugates. (If z is real, z
z, so z is its own conjugate.) We have seen that if we allow complex numbers as solutions, then every quadratic equation has a solution. More generally, it is true that every polynomial equation a n x n
a n1 x n1
  
a 1 x
a 0
0 of degree at least one has a solution among the complex numbers. This fact is known as the Fundamental Theorem of Algebra and was proved by Gauss. POLAR FORM

We know that any complex number z
a
bi can be considered as a point
a, b and that any such point can be represented by polar coordinates
r,   with r  0. In fact,

Im

a+bi r

a
r cos 

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b

b
r sin 

¨ 0

FIGURE 4

a

Re

as in Figure 4. Therefore, we have z
a
bi

r cos  

r sin  i

COMPLEX NUMBERS ■ 3

Thus, we can write any complex number z in the form z
r
cos 
i sin  

 

r
z
sa 2
b 2

where

tan 


and

b a

The angle  is called the argument of z and we write 
arg
z. Note that arg
z is not unique; any two arguments of z differ by an integer multiple of 2. EXAMPLE 4 Write the following numbers in polar form.

(a) z
1
i SOLUTION

(b) w
s3  i

 

(a) We have r
z
s12
12
s2 and tan 
1, so we can take 
 4. Therefore, the polar form is Im

1+i

z
s2

œ„ 2 π 4

0

_

π 6



cos

 
i sin 4 4



 

(b) Here we have r
w
s3
1
2 and tan 
1 s3 . Since w lies in the fourth quadrant, we take 
 6 and

Re

    

2

w
2 cos 

œ„ 3-i

 6


i sin 

 6

The numbers z and w are shown in Figure 5.

FIGURE 5

The polar form of complex numbers gives insight into multiplication and division. Let z1
r1
cos 1
i sin 1 

z2
r2
cos  2
i sin  2 

be two complex numbers written in polar form. Then Im

z™

z1 z2
r1r2
cos 1
i sin 1 
cos  2
i sin  2 

z¡


r1r2 
cos 1 cos  2  sin 1 sin  2 
i
sin 1 cos  2
cos 1 sin  2 

¨™

Therefore, using the addition formulas for cosine and sine, we have

¨¡ Re

¨¡+¨™

z1z2
r1r2 cos
1
 2 
i sin
1
 2 

1

z¡z™

This formula says that to multiply two complex numbers we multiply the moduli and add the arguments. (See Figure 6.) A similar argument using the subtraction formulas for sine and cosine shows that to divide two complex numbers we divide the moduli and subtract the arguments.

FIGURE 6

Im

z

z1 r1
cos
1   2 
i sin
1   2  z2 r2

r ¨ Thomson Brooks-Cole copyright 2007

0

_¨ 1 r

Re

In particular, taking z1
1 and z2
z, (and therefore 1
0 and  2
 ), we have the following, which is illustrated in Figure 7.

1 z

If FIGURE 7

z2  0

z
r
cos 
i sin  ,

then

1 1

cos   i sin  . z r

4 ■ COMPLEX NUMBERS

EXAMPLE 5 Find the product of the complex numbers 1
i and s3  i in polar form. SOLUTION From Example 4 we have

   

 
i sin 4 4

1
i
s2 cos s3  i
2 cos 

and

Im

z=1+i œ„ 2

zw

2œ„2

So, by Equation 1,

π 12

0

  


1
i(s3  i)
2s2 cos

Re

2


2s2 cos

w=œ„ 3-i FIGURE 8

 6



 


i sin 

   4 6

 6

  
i sin

   4 6



 
i sin 12 12

This is illustrated in Figure 8. Repeated use of Formula 1 shows how to compute powers of a complex number. If z
r
cos 
i sin   then

z 2
r 2
cos 2
i sin 2 

and

z 3
zz 2
r 3
cos 3
i sin 3 

In general, we obtain the following result, which is named after the French mathematician Abraham De Moivre (1667–1754). 2

De Moivre’s Theorem If z
r
cos 
i sin   and n is a positive integer, then

z n
r
cos 
i sin   n
r n
cos n
i sin n 

This says that to take the nth power of a complex number we take the nth power of the modulus and multiply the argument by n. EXAMPLE 6 Find SOLUTION Since

1 2

( 12
12 i)10.
12 i
12
1
i, it follows from Example 4(a) that 12
12 i has the polar

form 1 1 s2
i
2 2 2



cos

 
i sin 4 4



So by De Moivre’s Theorem,



   

1 1
i 2 2

10




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25 2 10

10

s2 2

cos

cos

10 10
i sin 4 4

5 5
i sin 2 2








1 i 32

De Moivre’s Theorem can also be used to find the n th roots of complex numbers. An n th root of the complex number z is a complex number w such that wn
z

COMPLEX NUMBERS ■ 5

Writing these two numbers in trigonometric form as w
s
cos
i sin 

and

z
r
cos 
i sin  

and using De Moivre’s Theorem, we get s n
cos n
i sin n 
r
cos 
i sin   The equality of these two complex numbers shows that sn
r and

cos n
cos 

s
r 1 n

or

sin n
sin 

and

From the fact that sine and cosine have period 2 it follows that n

2k

Thus

 

w
r 1 n cos





2k n

 


2k n

or


2k n


i sin



Since this expression gives a different value of w for k
0, 1, 2, . . . , n  1, we have the following. 3 Roots of a Complex Number Let z
r
cos 
i sin   and let n be a positive integer. Then z has the n distinct n th roots

 

wk
r 1 n cos


2k n

 
i sin




2k n

where k
0, 1, 2, . . . , n  1.

 

Notice that each of the nth roots of z has modulus wk
r 1 n. Thus, all the nth roots of z lie on the circle of radius r 1 n in the complex plane. Also, since the argument of each successive nth root exceeds the argument of the previous root by 2 n , we see that the n th roots of z are equally spaced on this circle. EXAMPLE 7 Find the six sixth roots of z
8 and graph these roots in the complex

plane. SOLUTION In trigonometric form, z
8
cos 
i sin  . Applying Equation 3 with n
6,

we get



wk
8 1 6 cos


2k 
2k
i sin 6 6



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We get the six sixth roots of 8 by taking k
0, 1, 2, 3, 4, 5 in this formula:

  

    

w0
8 1 6 cos

 
i sin 6 6

w1
8 1 6 cos

 
i sin 2 2

w2
8 1 6 cos

5 5
i sin 6 6


s2



1 s3
i 2 2


s2 i


s2 



1 s3
i 2 2

6 ■ COMPLEX NUMBERS

  

Im œ„2 i w¡ w™

2 _œ„

w¸ 0

w∞ _œ„2 i

7 7
i sin 6 6

w4
8 1 6 cos

3 3
i sin 2 2

w5
8 1 6 cos

11 11
i sin 6 6

œ„ 2 Re

w£ w¢

    

w3
8 1 6 cos


s2 



1 s3  i 2 2


s2 i
s2



1 s3  i 2 2

All these points lie on the circle of radius s2 as shown in Figure 9.

FIGURE 9

The six sixth roots of z=_8

COMPLEX EXPONENTIALS

We also need to give a meaning to the expression e z when z
x
iy is a complex number. The theory of infinite series as developed in Chapter 8 can be extended to the case 2 as our guide, where the terms are complex numbers. Using the Taylor series for e x (8.7.12) we define ez


4





n
0

zn z2 z3
1
z


 n! 2! 3!

and it turns out that this complex exponential function has the same properties as the real exponential function. In particular, it is true that e z
z
e z e z

5

1

2

1

2

If we put z
iy, where y is a real number, in Equation 4, and use the facts that i 2
1,

i 3
i 2i
i,

i 4
1, i 5
i, . . .

we get e iy
1
iy

1
iy 




1


iy2
iy3
iy4
iy5



 2! 3! 4! 5! y2 y3 y4 y5 i

i
 2! 3! 4! 5!

 



y2 y4 y6 y3 y5


i y 
  2! 4! 6! 3! 5!


cos y
i sin y Here we have used the Taylor series for cos y and sin y (Equations 8.7.17 and 8.7.16). The result is a famous formula called Euler’s formula:

6

e iy
cos y
i sin y

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Combining Euler’s formula with Equation 5, we get 7

e x
iy
e xe iy
e x
cos y
i sin y

COMPLEX NUMBERS ■ 7

(a) e i

EXAMPLE 8 Evaluate: ■ ■

(b) e1
i 2

SOLUTION

We could write the result of Example 8(a)

as

(a) From Euler’s equation (6) we have

e i
1
0

e i
cos 
i sin 
1
i
0
1

This equation relates the five most famous numbers in all of mathematics: 0, 1, e, i, and .

(b) Using Equation 7 we get



 
i sin 2 2

e1
i 2
e1 cos






1 i 0
i
1
e e

Finally, we note that Euler’s equation provides us with an easier method of proving De Moivre’s Theorem: r
cos 
i sin   n

re i n
r ne in
r n
cos n
i sin n 

EXERCISES A Click here for answers.

25–28

Click here for solutions.

S

5

3.
2
5i 
4  i

4.
1  2i 
8  3i 

5. 12
7i

6. 2i ( 12  i )

1
4i 7. 3
2i

3
2i 8. 1  4i

9.

27. 3
4i

2. (4  2 i)  (9
2 i) 1

1 1
i

■

■

11. i 3

12. i 100

13. s25

14. s3 s12

■

■

15–17

■

■

■

■

■

■

■

■

■

■

32. z
4(s3
i ),

w
3  3i

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

21. x
2x
5
0

22. 2x  2x
1
0

23. z
z
2
0

24. z
z

0

2

■

1 2

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

■

5

■

■

■

■

■

Find the indicated roots. Sketch the roots in the complex 38. The fifth roots of 32

■

■

■

■

40. The cube roots of 1
i ■

■

■

43. e i 3

44. e i

45. e 2
i

46. e 
i

■

■

■

■

■

■

■

■

■

■

Write the number in the form a
bi. 42. e 2 i

■

■

■

■

■

■

■

■

47. Use De Moivre’s Theorem with n
3 to express cos 3 and

1 4

■

■

36.
1  i 8

5

i 2

41. e

20. x
1

■

■

41–46

19. 4x
9
0

2

■

34. (1  s3 i )

33.
1
i 20

■

4

2

■

Find the indicated power using De Moivre’s Theorem.

39. The cube roots of i

Find all solutions of the equation.

2

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■

■

(c) z
z , where n is a positive integer [Hint: Write z
a
bi, w
c
di.] 2

■

plane.

n

n

■

■

37. The eighth roots of 1

■

■

w
1
s3 i

w
1
i

37–40

16. 1
2 s2 i

18. Prove the following properties of complex numbers. (a) z
w
z
w (b) zw
z w

■

■

31. z
2 s3  2i,

■

17. 4i

19–24

■

w
8i

35. (2 s3
2i )

number.

■

■

30. z
4 s3  4i,

33–36

Find the complex conjugate and the modulus of the

15. 12  5i

■

29. z
s3
i,

■

■

28. 8i ■

29–32 Find polar forms for z w, z w, and 1 z by first putting z and w into polar form.

3 4  3i

10.

26. 1  s3 i

25. 3
3i

1–14 Evaluate the expression and write your answer in the form a
bi. 1.
5  6i 

3
2i 

Write the number in polar form with argument between 0

and 2.

■

■

sin 3 in terms of cos  and sin .

8 ■ COMPLEX NUMBERS

48. Use Euler’s formula to prove the following formulas for cos x

and sin x : cos x


e ix
eix 2

sin x


e ix  eix 2i

49. If u
x
f
x
it
x is a complex-valued function of a real

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variable x and the real and imaginary parts f
x and t
x are differentiable functions of x, then the derivative of u is defined to be u
x
f 
x
it
x. Use this together with Equation 7 to prove that if F
x
e rx, then F
x
re rx when r
a
bi is a complex number.

50. (a) If u is a complex-valued function of a real variable, its

indefinite integral x u
x dx is an antiderivative of u. Evaluate

ye


1
i x

dx

(b) By considering the real and imaginary parts of the integral in part (a), evaluate the real integrals

ye

x

cos x dx

and

ye

x

sin x dx

(c) Compare with the method used in Example 4 in Section 6.1.

COMPLEX NUMBERS ■ 9

ANSWERS S

1. 8  4i 1 2

(2 s2 )cos
13 12
i sin
13 12, 14 cos
 6
i sin
 6

7.

 12 i

11 13




10 13

11. i

37. 1, i, (1 s2 )
1  i 

i 13. 5i

15. 12
5i; 13

3

23.  2  (s7 2)i

21. 1  2i

39. (s3 2)
2 i, i 1

Im

Im i

19.  2 i

17. 4i, 4

35. 512 s3
512i

33. 1024

3. 13
18i

5. 12  7i 9.

31. 4 s2 cos
7 12
i sin
7 12,

Click here for solutions.

1

0

1

0

Re

25. 3 s2 cos
3 4
i sin
3 4

[

27. 5{cos tan

1 4 3

29. 4cos
 2
i sin
 2, cos
 6
i sin
 6,

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1 2

_i

( )]
i sin[tan ( )]}

1 4 3

cos
 6
i sin
 6

Re

41. i

43.

1 2


(s3 2) i

45. e 2

47. cos 3
cos3  3 cos  sin2, sin 3
3 cos2 sin   sin3

10 ■ COMPLEX NUMBERS

SOLUTIONS

1. (5 − 6i) + (3 + 2i) = (5 + 3) + (−6 + 2)i = 8 + (−4)i = 8 − 4i

3. (2 + 5i)(4 − i) = 2(4) + 2(−i) + (5i)(4) + (5i)(−i) = 8 − 2i + 20i − 5i2 = 8 + 18i − 5(−1) = 8 + 18i + 5 = 13 + 18i

5. 12 + 7i = 12 − 7i

1 + 4i 3 − 2i 3 − 2i + 12i − 8(−1) 11 10 11 + 10i 1 + 4i = · = = + i = 3 + 2i 3 + 2i 3 − 2i 32 + 22 13 13 13 1 1−i 1−i 1−i 1 1 1 = · = = = − i 9. 1+i 1+i 1−i 1 − (−1) 2 2 2 7.

11. i3 = i2 · i = (−1)i = −i √ √ 13. −25 = 25 i = 5i

s √ √ 122 + (−5)2 = 144 + 25 = 169 = 13 t √ 17. −4i = 0 − 4i = 0 + 4i = 4i and |−4i| = 02 + (−4)2 = 16 = 4 t t 19. 4x2 + 9 = 0 ⇔ 4x2 = −9 ⇔ x2 = − 94 ⇔ x = ± − 94 = ± 94 i = ± 32 i.

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15. 12 − 5i = 12 + 15i and |12 − 15i| =

21. By the quadratic formula, x2 + 2x + 5 = 0 ⇔ s √ −2 ± 22 − 4(1)(5) −2 ± −16 −2 ± 4i x= = = = −1 ± 2i. 2(1) 2 2 s √ √ −1 ± 12 − 4(1)(2) −1 ± −7 1 7 2 23. By the quadratic formula, z + z + 2 = 0 ⇔ z = = =− ± i. 2(1) 2 2 2 s √ 3 25. For z = −3 + 3i, r = (−3)2 + 32 = 3 2 and tan θ = −3 = −1 ⇒ θ = 3π 4 (since z lies in the second √   3π 3π quadrant). Therefore, −3 + 3i = 3 2 cos 4 + i sin 4 . √   27. For z = 3 + 4i, r = 32 + 42 = 5 and tan θ = 43 ⇒ θ = tan−1 43 (since z lies in the first quadrant).      Therefore, 3 + 4i = 5 cos tan−1 43 + i sin tan−1 43 . t√  √   2 29. For z = 3 + i, r = 3 + 12 = 2 and tan θ = √13 ⇒ θ = π6 ⇒ z = 2 cos π6 + i sin π6 . √ √   For w = 1 + 3 i, r = 2 and tan θ = 3 ⇒ θ = π3 ⇒ w = 2 cos π3 + i sin π3 . Therefore,        zw = 2 · 2 cos π6 + π3 + i sin π6 + π3 = 4 cos π2 + i sin π2 ,          z/w = 22 cos π6 − π3 + i sin π6 − π3 = cos − π6 + i sin − π6 , and 1 = 1 + 0i = 1(cos 0 + i sin 0) ⇒           1/z = 12 cos 0 − π6 + i sin 0 − π6 = 12 cos − π6 + i sin − π6 . For 1/z, we could also use the formula that   precedes Example 5 to obtain 1/z = 12 cos π6 − i sin π6 . t √  √ 2 √ = − √1 2 3 + (−2)2 = 4 and tan θ = 2−2 31. For z = 2 3 − 2i, r = 3 3 √      ⇒ θ = − π6 ⇒ z = 4 cos − π6 + i sin − π6 . For w = −1 + i, r = 2, √   1 3π tan θ = −1 . Therefore, = −1 ⇒ θ = 3π ⇒ w = 2 cos 3π 4 4 + i sin 4 √   π √    π   3π 3π 7π zw = 4 2 cos − 6 + 4 + i sin − 6 + 4 = 4 2 cos 12 + i sin 7π , 12   π   π    11π   11π  4 3π 3π 4 z/w = √2 cos − 6 − 4 + i sin − 6 − 4 = √2 cos − 12 + i sin − 12 √   13π 13π = 2 2 cos 12 + i sin 12 , and        1/z = 14 cos − π6 − i sin − π6 = 14 cos π6 + i sin π6 . √  √  33. For z = 1 + i, r = 2 and tan θ = 11 = 1 ⇒ θ = π4 ⇒ z = 2 cos π4 + i sin π4 . So by De Moivre’s Theorem, k√  l20  1/2 20   cos 204· π + i sin 204· π 2 cos π4 + i sin π4 = 2 (1 + i)20 = = 210 (cos 5π + i sin 5π) = 210 [−1 + i(0)] = −210 = −1024

COMPLEX NUMBERS ■ 11

t √  √ √ 2 2 2 3 + 22 = 16 = 4 and tan θ = 2√ 35. For z = 2 3 + 2i, r = = √13 ⇒ θ = π6 ⇒ 3   z = 4 cos π6 + i sin π6 . So by De Moivre’s Theorem,  √ k √ 5   l √ 5   2 3 + 2i = 4 cos π6 + i sin π6 = 1024 − 23 + 12 i = −512 3 + 512i = 45 cos 5π + i sin 5π 6 6

37. 1 = 1 + 0i = 1 (cos 0 + i sin 0). Using Equation 3 with r = 1, n = 8, and θ = 0, we have      kπ kπ 0 + 2kπ 0 + 2kπ 1/8 cos wk = 1 + i sin = cos + i sin , where k = 0, 1, 2, . . . , 7. 8 8 4 4   w0 = 1(cos 0 + i sin 0) = 1, w1 = 1 cos π4 + i sin π4 = √12 + √12 i,     = − √12 + √12 i, w2 = 1 cos π2 + i sin π2 = i, w3 = 1 cos 3π + i sin 3π 4 4   5π w4 = 1(cos π + i sin π) = −1, w5 = 1 cos 5π = − √12 − √12 i, 4 + i sin 4     = −i, w7 = 1 cos 7π = √12 − √12 i w6 = 1 cos 3π + i sin 3π + i sin 7π 2 2 4 4   39. i = 0 + i = 1 cos π2 + i sin π2 . Using Equation 3 with r = 1, n = 3, and θ =  π  π  + 2kπ + 2kπ wk = 11/3 cos 2 + i sin 2 , where k = 0, 1, 2. 3 3  √  w0 = cos π6 + i sin π6 = 23 + 12 i √   5π = − 23 + 12 i w1 = cos 5π 6 + i sin 6   9π = −i w2 = cos 9π 6 + i sin 6 41. Using Euler’s formula (6) with y =

π , 2

we have

π , 2

we have eiπ/2 = cos π2 + i sin π2 = 0 + 1i = i. √ π π 1 π 3 43. Using Euler’s formula (6) with y = , we have eiπ/3 = cos + i sin = + i. 3 3 3 2 2 45. Using Equation 7 with x = 2 and y = π, we have e2+iπ = e2 eiπ = e2 (cos π + i sin π) = e2 (−1 + 0) = −e2 .

47. Take r = 1 and n = 3 in De Moivre’s Theorem to get

[1(cos θ + i sin θ)]3 = 13 (cos 3θ + i sin 3θ) (cos θ + i sin θ)3 = cos 3θ + i sin 3θ   cos3 θ + 3 cos2 θ (i sin θ) + 3(cos θ)(i sin θ)2 + (i sin θ)3 = cos 3θ + i sin 3θ     cos3 θ + 3 cos2 θ sin θ i − 3 cos θ sin2 θ − sin3 θ i = cos 3θ + i sin 3θ  3    cos θ − 3 sin2 θ cos θ + 3 sin θ cos2 θ − sin3 θ i = cos 3θ + i sin 3θ

Equating real and imaginary parts gives cos 3θ = cos3 θ − 3 sin2 θ cos θ

and sin 3θ = 3 sin θ cos2 θ − sin3 θ

49. F (x) = erx = e(a+bi)x = eax+bxi = eax (cos bx + i sin bx) = eax cos bx + i(eax sin bx) ⇒ F 0 (x) = (eax cos bx)0 + i(eax sin bx)0

= (aeax cos bx − beax sin bx) + i(aeax sin bx + beax cos bx) = a [eax (cos bx + i sin bx)] + b [eax (− sin bx + i cos bx)]    = aerx + b eax i2 sin bx + i cos bx

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= aerx + bi [eax (cos bx + i sin bx)] = aerx + bierx = (a + bi)erx = rerx