Complex Analysis

“ca: chapter_03” — 2012/3/27 — 20:49 — page 95 — #8

Elementary Functions

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95

3.6.3 Zeros of sin z, cos z

A zero of a function f ( z) is a number z

0

such that f (z

0

)=0.

We know that sin z = sin

(

x + iy

)

= sin x cos iy + cos x sin iy = sin x cosh y + i cos x sinh y

⇔|sin z|

2

= sin

2

x cosh

2

y + cos

2

x sinh

2

y

= sin

2

x(1 + sinh

2

y ) + (1 − sin

2

x) sinh

2

y

= sin

2

x + sinh

2

y (3.16)

∴ sin z =0⇔sin x = 0, sinh y =0, x, y being real

⇔ x = nπ , y = 0, where n ∈ I

Thus, the zeros of sin z are given by z = nπ , where n ∈ I.

Similarly, we can obtain cos z = cos x cosh y − i sin x sinh y

⇔|cos z|

2

= cos

2

x + sinh

2

y

∴ cos z = 0 ⇔ cos x = 0, sinh y = 0

⇔ x = (2n + 1)

π

2

, y = 0, wheren