Page 30 - Calculus with Complex Numbers

2 . I I T he other trigonom etric functions
W e define tan z cot z sec z cosec z in terms of sin z cos z as follows.

sin z cos z 1
tan z = , cot z = sec z =
cos z sin z cos z
Similarly for the other hyperbolic functions.
These functions all have singularities. For example tan z has singularities
at the zeros of cos z, that is z = n:r + >/2. The corresponding hyperbolic
function tanh z = sinh z/ cosh z has singularities at the zeros of cosh z, that is
z = i trlzr + >/2).

u
2 . I 2 T h e I ogar i th m i c f n cti o n
The graph of y = log .x for real .x is as in Figtlre 2.9. Observe that log .x is only
delined for .x > 0. This is because the real exponential function only takes positive
values (see Figtlre 2.4).
To define log z for complex z we use the polar form z = reio . W e get

since arg c is many valued it follows that log z is also many valued. W e define
tjm yrinciyal value of log c to be the one obtained by taking the principal value of
arg c. For example, we have 1 + i = .x ein'/'î (PV) therefore

Observethat log z has a singularity at z = 0 sincewe cannotdeline log r for r = 0.
To get the complex graph for w = log z it is best to consider the action of
log z on the circles Iz I= constant and the half lines arg z = constant in the z-

plane. These transform to the grid lines Re w = constant lm w = corlstant in the
w-plane (see Figure 2. 10).

Ffgure 2.9