Complex Linear Functions
The function is
w
=
az+b
. The blue grid is in the
z
=
x
+
iy
domain, the red image is in the
w
=
u
+
iv
domain.
We will consider
w
= (1-
i
)
z
+(2+
i
).
Along the real axis, think how
w = az+b
is
(1-
i
)
t
+ (2+
i
) = [
t
+2, -
t
+1]
Which yields the path
y
= -
x
+ 3.
However, along the imaginary axis,
w = az+b
is
(1-
i
)
it
+ (2+
i
) = [
t
+2,
t
+1]
Which yields the path
y
=
x
- 1.