3. ELEMENTARY
FUNCTIONS



Exponential and Trigonometric Functions



If z = x + iy, we define the exponential function exp z = e^{x} cis y (= e^{z}).
Notes
1. We can give an alternative definition in terms of power series.
Writing out a formal series for e ^{iy} gives cis y.
2. If y = 0, then exp z = exp x = e^{x}. Thus the complex exponential function naturally extends the real function.
3. In this definition, y is in radian measure.


Properties of the Exponential (II)



If y is a real number, we have
exp(iy) = cos y + i sin y,
exp(–iy) = cos y – i sin y,
cos y = ^{1}/_{2}. (exp(iy) + exp(–iy)),
sin y = ^{1}/_{2}_{i}.(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}/_{2}_{i}.(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.


Properties of Sine and Cosine



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. sin^{2 }z + cos^{2 }z = 1.
4. sin(z_{1} + z_{2}) = sin z_{1 }cos z_{2 }+ sin z_{2 }cos z_{1} 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 multivalued. Let us write
w = log z z = exp w.
If we set z = r cis , w = u + iv, then r cis = e^{u} cis v.
From this, we deduce that
r = e^{u},^{ }u = ln r, v = + 2k.
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.



1. A path which crosses the cut moves to the next branch.

2. If z is real and positive, then Log z = ln r.

3. We can think of the branch planes interleaved together, with the xaxis as a common axis. A path drawn about the origin in one branch plane reaches the cut and then passes to the next branch plane.

4. Our choice of the positive xaxis for the cut was somewhat arbitrary. Other branch cuts are possible; but O is common to them all – O is a branch point.



Properties of the Logarithm (I)



These functions are continuous on the given domain and satisfy the CauchyRiemann 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 CauchyRiemann equations.]


Properties of the Logarithm (II)



All branches have the same derivative, since they differ by a constant.
3. Inverse Property
exp(log z) = z (for any branch)
log(exp z) = z (for a particular branch).
log z_{1 }+ log z_{2} = log(z_{1} . z_{2})
log z_{1 }– log z_{2} = log(z_{1 }/ z_{2})
providing we choose the appropriate logarithm branch on the right.


Examples on the Logarithm



Example 1. Evaluate Log(–1) + Log(–1).
Now –1 = 1 . cis , so Log(–1) = 0 + i.
Hence 2Log(–1) = 2i = log 1, but not Log 1 (= 0).

Using our knowledge of real powers, we define the complex power z^{c} (c C) by
z^{c }= exp(c log z), (z 0).
Since z^{c} is defined in terms of the logarithm, we expect z^{c} to be multivalued, so we use the cut plane as for the logarithm. Then since log z is singlevalued and analytic in the cut plane, so is z^{c}. Now
