Page 707 - Advanced_Engineering_Mathematics o'neil

19.3 The Exponential and Trigonometric Functions 687

where
1 y −y 1 y −y
cosh(y) = (e + e ) and sinh(y) = (e − e ).
2 2
To verify these, write

1 iz −iz
cos(z) = e + e
2
1 i(x+iy) −i(x+iy)
= e + e
2
1 ix −y −ix y
= e e + e e
2
1 −y y
= e (cos(x) + i sin(x)) + e (cos(x) − i sin(x))
2
1 i
y
= cos(x) e + e −y + sin(x) e −y − e y
2 2
= cos(x)cosh(y) − i sin(x)sinh(y).
Similarly,
sin(z) = sin(x)cosh(y) + i cos(x)sinh(y).

If z = x is real, then y =0, the complex sine agrees with the real sine, and the complex cosine
agrees with the real cosine. In this sense, sin(z) and cos(z) are extensions of sin(x) and cos(x)
to the complex plane.
A multiplication shows that, for all z,
2
2
cos (z) + sin (z) = 1,
as we should expect. Other identities for the real sine and cosine functions extend readily to their
complex counterparts, although their derivations are very much simplified in the complex case
because exponential functions are easy to compute with. For example, suppose we want to show
that sin(2z) = 2sin(z)cos(z). Compute
1 iz −iz iz −iz
2sin(z)cos(z) = e − e e + e
2i
1 2iz −2iz
= e − e = sin(2z).
2i
From the Cauchy-Riemann equations, cos(z) and sin(z) are differentiable for all z. Further-
more, using equations (19.1),
d ∂u ∂v
cos(z) = + i
dz ∂x ∂y
=−sin(x)cosh(y) − i cos(x)sinh(y) =−sin(z),
and similarly,
d
sin(z) = cos(z).
dz
The complex sine and cosine functions exhibit some properties that are not seen in the real
case. For example, the real sine and cosine functions are bounded: |cos(x)|≤ 1 and |sin(x)|≤ 1
for all real x. But the complex sine and cosine are not bounded functions in the complex plane.
For real y,
1
y
cos(iy) = (e −y + e ),
2

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 15, 2010 18:5 THM/NEIL Page-687 27410_19_ch19_p667-694