Page 31 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes So, suppose z = x + 0i = x. Then,
e = cos x + i sin x, and
ix
e = cos x – i sin x.
–ix
Thus,

ix
e + e –ix
cos x = ,
2
e – e –ix
ix
sin x =
2i
Next, observe that the sine and cosine functions are entire–they are simply linear combinations
of the entire functions e and e . Moreover, we see that
iz
–iz
d cosz,and d sinz,
dz sinz  dz cosz  

just as we would hope.
It may not have been clear to you back in elementary calculus what the so-called hyperbolic sine
and cosine functions had to do with the ordinary sine and cosine functions.

Now perhaps it will be evident. Recall that for real t,
t
t
e  e t  e  e t 
sin ht  , and cos h t 
2 2
Thus,
e i(it)  e  i(it) e – e t 
t
sin (it)    isin h t
2i 2
Similarly,

cos (it) = cos ht.
Most of the identities you learned in the 3rd grade for the real sine and cosine functions are also
valid in the general complex case. Let’s look at some.

1
iz
sin z + cos z =  (e  e  iz 2 (e  e  iz 2 
iz
2
) 
) 
2

4
1 2iz iz 2iz 2iz iz 2iz
iz 
iz 
= 4  e  2e e  e   e  2e e  e   

1
= (2  2) 1

4
It is also relative straight-forward and easy to show that:
sin(z ± w) = sin z cos w ± cos z sin w, and
cos(z ± w) = cos z cos w  sin z sin w

24 LOVELY PROFESSIONAL UNIVERSITY