ph
squaresmallsolid
z
squaresmallsolid
z
Figure 7.18.
Complex Curve and Tangent.
notation
x
(
t
) = (
x
(
t
)
,y
(
t
) ) to complex notation
z
(
t
) =
x
(
t
)+ i
y
(
t
). All the usual vectorial
curve terminology — closed, simple (nonself intersecting), piecewise smooth, etc.
— is
employed without modification.
In particular, the
tangent vector
to the curve can be
identified as the complex number
squaresmallsolid
z
(
t
) =
squaresmallsolid
x
(
t
) + i
squaresmallsolid
y
(
t
), where we use dots to indicated
derivatives with respect to the parameter
t
. Smoothness of the curve is guaranteed by the
requirement that
squaresmallsolid
z
(
t
)
negationslash
= 0.
Example 7.28.
(
a
) The curve
z
(
t
) =
e
i
t
= cos
t
+ i sin
t,
for
0
≤
t
≤
2
π,
parametrizes the unit circle

z

= 1 in the complex plane.
Its complex tangent
squaresmallsolid
z
(
t
) =
i
e
i
t
= i
z
(
t
) is obtained by rotating
z
through 90
◦
.
(
b
) The complex curve
z
(
t
) = cosh
t
+ i sinh
t
=
1 + i
2
e
t
+
1
−
i
2
e
−
t
,
−∞
<t<
∞
,
parametrizes the right hand branch of the hyperbola Re
z
2
=
x
2
−
y
2
= 1. The complex
tangent vector is
squaresmallsolid
z
(
t
) = sinh
t
+ i cosh
t
= i
z
(
t
).
When we interpret the curve as the motion of a particle in the complex plane, so that
z
(
t
) is the position of the particle at time
t
, the tangent
squaresmallsolid
z
(
t
) represents its instantaneous
velocity.
The modulus of the tangent,

squaresmallsolid
z

=
radicalbig
squaresmallsolid
x
2
+
squaresmallsolid
y
2
, indicates the particle’s speed,
while its phase ph
squaresmallsolid
z
measures the direction of motion, as prescribed by the angle that the
curve makes with the horizontal; see Figure 7.18.
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 Fall '10
 Olver
 Differential Equations, Equations, Partial Differential Equations, Analytic function, Holomorphic function, Logarithm, Peter J. Olver

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