If z = x + iy, we define the exponential function exp z = ex cis y (= ez).
1. We can give an alternative definition in terms of power series.
2. If y = 0, then exp z = exp x = ex. Thus the complex exponential function naturally extends the real function.
3. In this definition, y is in radian measure.
8. We observe that exp(z + 2i) = exp z.exp(2i),
and that exp(2i) = e0.(cos 2 + i sin 2) = 1.
It follows that exp(z + 2i) = exp z.
Thus we can divide the z-plane into periodic strips. Each strip in the z-plane is mapped to the whole w-plane excluding the origin.
We note the further two properties of the exponential:
8. exp = .
If y is a real number, we have
exp(iy) = cos y + i sin y,
cos y = 1/2. (exp(iy) + exp(iy)),
sin y = 1/2i.(exp(iy) exp(iy)).
Thus it is natural to define cosine and sine as:
cos z = 1/2. (exp(iz) + exp(iz)),
sin z = 1/2i.(exp(iz) exp(iz)).
These are Euler's relations. Again notice here how we try to generalize, or extend, a real situation to the complex case.
1. Both functions are entire:
(sin z) = cos z, (cos z) = sin z.
2. Both functions are periodic, of period 2. This follows from the periodicity of the exponential function.
The functions satisfy the usual identities, as in the real case.
3. sin2 z + cos2 z = 1.
4. sin(z1 + z2) = sin z1 cos z2 + sin z2 cos z1 etc.
5. sin( z) = sin z, cos( z) = cos z etc.
Does the exponential function have an inverse logarithmic function? Since the exponential function is periodic, any inverse would have to be multi-valued. Let us write
w = log z z = exp w.
If we set z = r cis , w = u + iv, then r cis = eu cis v.
From this, we deduce that
r = eu, u = ln r, v = + 2k.
w = log z = ln | z | + i( + 2k) (k Z).
Thus there are infinitely many values of log z, the different values differing by 2ki. Each value of k gives a branch of the logarithm.
Consider Log z = ln r + i ( < , r > 0) that is, over the open domain excluding the cut. There are difficulties on the cut, for is not continuous there for any branch. Hence, for example, the Log function is not continuous on the cut, and so the Log function is not differentiable there.
1. Log z is analytic over the open domain ( < < , r > 0).
Writing Log z = u + iv, we have u = 1/2 ln (x2 + y2), v = = arctan y/x.
These functions are continuous on the given domain and satisfy the Cauchy-Riemann equations there. Hence by Theorem 2.5, Log z is analytic.
[Note There is a problem in defining arctan here when x = 0. We could overcome this by defining = arccot x/y, or by taking time to develop a polar form of the Cauchy-Riemann equations.]
All branches have the same derivative, since they differ by a constant.
3. Inverse Property
exp(log z) = z (for any branch)
4. Sums and Differences
log z1 + log z2 = log(z1 . z2)
providing we choose the appropriate logarithm branch on the right.
Using our knowledge of real powers, we define the complex power zc (c C) by
zc = exp(c log z), (z 0).
Since zc is defined in terms of the logarithm, we expect zc to be multivalued, so we use the cut plane as for the logarithm. Then since log z is single-valued and analytic in the cut plane, so is zc. Now
1. i 1/4 = exp(1/4 log i) = exp(1/4 i (/2 2k)) = exp(i/8 ki/2) four values.
2. i i = exp(i log i) = exp(i (/2 2k) i) = exp(/2 2k).
3. What is the relationship between exp z and ez ?
Clearly ez = exp(z log e). Now e = e cis 0, so log e = 1 2ki, and
It follows that ez = exp z . exp(2kiz). Setting k = 0 gives ez = exp z.