Page 32 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 3: Elementary Functions

Other familiar ones follow from these in the usual elementary school trigonometry fashion. Notes
Let’s find the real and imaginary parts of these functions:
sin z = sin(x + iy) = sin x cos(iy) + cos x sin (iy)
= sin x cos hy + i cos x sin hy.
In the same way, we get cos z = cos x cos h y – i sin x sin hy.

3.3 Logarithms and Complex Exponents

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 thus many log z’s; one for each argument of z. The difference between any two of these
is, thus, an integral multiple of 2i. First, for any value of log z we have
= e
e log z = e ln|z|+ i arg z ln|z| i arg z = z.
e
This is familiar. But next there is a slight complication:
log(e ) = ln e + i arg e = x + y(y + 2k)i
x
z
z
= z + 2ki,
where k is an integer. We also have
log(zw) = ln(|z||w|) + i arg(zw)
= ln|z| + i arg z + ln|w| + i arg w + 2ki
= log z + log w + 2ki
for some integer k.

There is defined a function, called the principal logarithm, or principal branch of the logarithm,
function, given by
Log z = ln|z| + iArg z,

where Arg z is the principal argument of z. Observe that for any log z, it is true that log z =
Log z + 2ki for some integer k which depends on z. This new function is an extension of the real
logarithm function:
Log x = ln x + iArg x = ln x.
This function is analytic at a lot of places. First, note that it is not defined at z = 0, and is not
continuous anywhere on the negative real axis (z = x + 0i, where x < 0). So, let’s suppose
z = x + iy , where z is not zero or on the negative real axis, and see about a derivative of
0
0
0
0
Log z :
Log z Log z Log z Log z


lim 0  lim 0

z 0 z z z 0 z 0 z e log z  e log 0 z

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