Page 13 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes We shall see later as the drama of the term unfolds that this very suggestive notation is an
excellent choice. Now, we have in polar form
z = re ,
i
where r = |z| and  is any argument of z. Observe we have just shown that
e e = e i(+) .
i
i
It follows from this that e e = 1. Thus,
i
–i
1 e i  
e i 
It is easy to see that
z re i r ))

w  se i  s (cos(   ) isin(  

1.3 Summary

The modulus of a complex number z = x + iy is defined to be the nonnegative real number

2
2
x  y , which is, of course, the length of the vector interpretation of z.
The conjugate z of a complex number z = x + iy is defined by z = x – iy.

In other words, the conjugate of the sum is the sum of the conjugates. It is also true that

zw  z w. If z = x + iy, then x is called the real part of z, and y is called the imaginary part
of z. These are usually denoted Re z and Im z, respectively. Observe then that z + z = 2 Rez
and z – z = 2 Imz.
Now, for any two complex numbers z and w consider


2
|z + w| = (z  w)(z  w) (z  w)(z  w)

= zz (wz  wz) ww
2
=|z|  2Re(wz) |w| 2

 |z| + 2|z||w| + |w| = (|z| + |w|) 2
2
2
In other words,
|z + w|  |z| + |w|
the so-called triangle inequality. (This inequality is an obvious geometric fact–can you
guess why it is called the triangle inequality?)
We shall see later as the drama of the term unfolds that this very suggestive notation is an

excellent choice. Now, we have in polar form
z = re ,
i
where r = |z| and  is any argument of z. Observe we have just shown that

e e = e i(+) .
i
i
It follows from this that e e = 1. Thus
–i
i
1 e i  
e i 

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