Page 34 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 3: Elementary Functions

We, thus, use the quite reasonable notation e = exp(z) and observe that we have extended Notes
z
the real exponential e to the complex numbers.
x
First, let’s verify that these are honest-to-goodness extensions of the familiar real functions,

cosine and sine–otherwise we have chosen very bad names for these complex functions.
So, suppose z = x + 0i = x. Then,
e = cos x + i sin x, and
ix
e = cos x – i sin x.
–ix
Thus,

e + e –ix
ix
cos x = ,
2
e – e –ix
ix
sin x =
2i
In the case of real functions, the logarithm function was simply the inverse of the exponential

function. Life is more complicated in the complex case—as we have seen, the complex
exponential function is not invertible.
There are many solutions to the equation e = w.
z
If z  0, we define log z by
log z = ln|z| + i arg z.
There are many values of log z, and so there can be many values of z . As one might guess,
c

e cLog z is called the principal value of z . c
Note that we are faced with two different definitions of z in case c is an integer. Let’s see
c
if we have anything to unlearn. Suppose c is simply an integer, c = n. Then
z = e n log z = e n(Log z + 2ki)
n
e
= e nLog z 2kni = e nLog z
There is, thus, just one value of z , and it is exactly what it should be: e nLog z = |z| e . It
n
n in arg z
is easy to verify that in case c is a rational number, z is also exactly what it should be.
c
3.5 Keywords

Exponential function: Let the so-called exponential function exp be defined by exp(z) = ex(cos y
+ i sin y),

Logarithm function: The logarithm function was simply the inverse of the exponential function.
Principal value: There are many values of log z, and so there can be many values of z . As one
c
might guess, e cLog z is called the principal value of z . c
3.6 Self Assessment

1. Let the so-called exponential function exp be defined by ...................
2. If z  0, we define log z by ...................

LOVELY PROFESSIONAL UNIVERSITY 27