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13.6 Complex Fourier Series 457

9. f has the graph of Figure 13.19. 11. f has the graph of Figure 13.21.

y
y
1
2
x
–3 –2 –1 1 2 3
1
–1
x
–3 –1 1 3

FIGURE 13.19 f (x) in Problem 9,
Section 13.5.
FIGURE 13.21 f (x) in Problem 11,
Section 13.5.

12. f has the graph of Figure 13.22.

10. f has the graph of Figure 13.20. y

y
k
k

x x
–2 2 4 –2 –1 0 1 2 3

FIGURE 13.20 f (x) in Problem 10, FIGURE 13.22 f (x) in Problem 12,
Section 13.5. Section 13.5.

13.6 Complex Fourier Series

There is a complex form of Fourier series that is sometimes used. As preparation for this, recall
that, in polar coordinates, a complex number (point in the plane) can be written
z = x + iy =r cos(θ) + ir sin(θ)
where

2
r =|z|= x + y 2
and θ is an argument of z. This is the angle (in radians) between the positive x− axis and the
line from the origin through (x, y), or this angle plus any integer multiple of 2π. Using Euler’s
formula, we obtain the polar form of z:
iθ
z =r[cos(θ) + i sin(θ)]=re .
Now
iθ
e = cos(θ) + i sin(θ),

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October 14, 2010 14:57 THM/NEIL Page-457 27410_13_ch13_p425-464