The matrix
1 1
i Ài
is non-singular, and so the inverse matrix exists (see Sect.
11.3) and (x y) can be described as
x y
ð Þ ¼
1
2
z z
Ã
ð
Þ
1 Ài
1 i
ð6:42Þ
or with individual components we have
x ¼
1
2
z þ z
Ã
ð
Þ and y ¼ À
i
2
z À z
Ã
ð
Þ,
ð6:43Þ
which can be understood as the “vector synthesis” (see Fig. 6.9).
Equations (6.41) and (6.42) imply that two sets of variables (x y) and (z z
à ) can
be dealt with on an equal basis. According to the custom, however, we prioritize the
notation using (x y) over that of (z z
Ã
). Hence, the complex function f is describe as
f x, y
ð Þ ¼ u x, y
ð Þþiv x, y
ð Þ,
ð6:44Þ
where both u(x, y) and v(x, y) are real functions of real variables x and y.
0
∗
−
−
∗
−
∗
− +
∗
+
∗
− −
∗
Fig. 6.9 Vector synthesis of z and z
Ã
x
0
(a)
z
1
i
0
(b)
Fig. 6.10 Ways a number (real or complex) is approached. (a) Two ways a number x 0 is
approached in a real number line. (b) Various ways a number z 0 is approached in a complex
plane. Here, only four ways are indicated
200
6 Theory of Analytic Functions