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Exponential and trigonometric functions
tion as a complex function From the first principles, we define the complex exponential func-
f (^) ( z ) that satisfies the following defining
properties:
f (^) ( z ) is entire,
f (^) ′ ( z ) =
f (^) ( z ),
f (^) ( x ) =
e x ,
x
is real.
Let
f (^) ( z ) =
u ( x, y
(^) iv
x, y
z
=
x (^) +
(^) iy
From property (1),
u
and
v
satisfy the Cauchy-Riemann relations. Combining (1) and (2)
u x (^) +
(^) iv
x
v y − (^) iu
y
u (^) +
(^) iv.
First, we observe that
u x =
u
and
v x =
v
and so
u
=
(^) e x g ( y )
and
v = e x h ( y ) ,
where
g ( y ) and
h ( y ) are arbitrary functions in
y .
1
We also have
v y =
u
and
u y =
v,
from which we deduce that the arbitrary functions are related by
h
(′ y ) =
g ( y )
and
(^) g
(′ y ) =
h ( y ) .
By eliminating
g ( y ) in the above relations, we obtain
h ′′ ( y ) =
− h ( y ).
The general solution of the above equation is given by
h ( y ) =
A
(^) cos
(^) y
(^) B
(^) sin
(^) y,
where
A
and
B
are arbitrary constants.
Furthermore, using
g ( y ) =
h ′ ( y ), we have
g ( y ) =
A
(^) sin
(^) y (^) +
B
(^) cos
(^) y.
2
The modulus of
e z
is non-zero since
| e z | = e x 6
for all
z
in
C
and so
e z
6 = 0 for all
z
in the complex
z -plane.
The range of the
Periodic propertyzero value.complex exponential function is the entire complex plane except the
e z
kπi
e z ,
for any
z
and integer
k ,
that is,
e z is periodic with the fundamental period 2
πi
. The complex
exponential function is periodic while its real counterpart is not.
4
period 2 Since the complex exponential function is periodic with fundamentalMapping properties of the complex exponential function
πi
, it is a many-to-one function. If we restrict
z
to lie within
the infinite strip
π <
Im
z
≤
π , then the mapping
w
=
e z
becomes
The vertical lineone-to-one.
x
=
(^) α
is mapped onto the circle
| w | = e α
, while the
horizontal line
y
=
β
is mapped onto the ray Arg
w
=
β .
When the vertical line
x
=
(^) α
moves further to the left, the mapped
circle
| w | = e α
shrinks to a smaller radius. When the horizontal line
in the
z -plane moves vertically from
y
=
π
to
y
=
π , the image ray
in the
w -plane traverses in anticlockwise sense from Arg
(^) w
π
to
Arg
w
=
(^) π
.
5
Consider the following function:Example
f (^) ( z ) =
e z α ,^
α
is real
Show that
f (^) ( z ) | is constant on the circle
x 2
(^) y
2 − (^) ax
a
is a
Write the equation of the circle as Solutionreal constant.
( x (^) −
(^2) a ) 2
(^) y 2
=
( (^2) a^ ) 2
which reveals that the circle is centered at
( 2 a , (^0) )
and has radius
(^2) a .
A possible parametric representation of the circle is
x
=
2 a (1 + cos
(^) θ )
and
y
=
(^2) a
sin
(^) θ,
π < θ
(^) π.
7
The parameter
θ
is the angle between the positive
x -axis and the
line joining the center
( (^2) a , (^0) )
to the point (
x, y
The complex rep-
resentation of the circle can be expressed as
z
=
(^2) a
(1 +
(^) e iθ ) ,
π < θ
π.
The modulus of
f (^) ( z ) when
z
lies on the circle is found to be
f (^) ( z ) | =
∣∣∣∣ ∣ e
2 α
a (1+
e iθ ) ∣∣∣∣ ∣ =
∣∣e∣∣ ∣ 2 a α
1+cos
(^) θ − i sin
(^) θ
2(1+cos
(^) θ )
∣∣∣∣ ∣
e aα
.^
θ The modulus value is equal to a constant with no dependence on , that is, independent of the choice of the point on the circle.
8
Let
z
x (^) +
(^) iy
, then
e iz
=
e − y (cos
(^) x (^) +
(^) i (^) sin
(^) x )
and
e − iz
=
e y (cos
(^) x (^) −
(^) i (^) sin
(^) x ) .
The complex sine and cosine functions are seen to be
sin
(^) z
sin
(^) x (^) cosh
(^) y
(^) i (^) cos
(^) x (^) sinh
(^) y,
cos
(^) z
cos
(^) x (^) cosh
(^) y (^) −
(^) i (^) sin
(^) x (^) sinh
(^) y.
Moreover, their moduli are found to be
(^) sin
(^) z | =
√
sin
2 x^ (^) + sinh
2 y,^
(^) cos
(^) z | =
√
cos
2 x^ (^) + sinh
2 y.^
Since sinh
(^) y
is unbounded at large values of
y , the above modulus
values can increase (as
y
does) without bound.
− While the real sine and cosine functions are always bounded between
1 and 1, their complex counterparts are unbounded.
10
The complex hyperbolic functions are defined by
sinh
(^) z
=
e z − (^) e − z
cosh
(^) z
=
e z
(^) e − z
tanh
(^) z
=
sinh
(^) z
cosh
(^) z .
The other hyperbolic functions cosech
z ,
sech
z
and coth
z
are
defined as the reciprocal of sinh
(^) z,
cosh
(^) z
and tanh
(^) z , respectively.
metric functions. SupposeIn fact, the hyperbolic functions are closely related to the trigono-
z
is replaced by
iz
, we obtain
sinh
(^) iz
i (^) sin
(^) z.
Similarly, one can show that
sin
(^) iz
i (^) sinh
(^) z,
cosh
(^) iz
= cos
(^) z,
cos
(^) iz
= cosh
(^) z.
11
A zero
α
of a function
f (^) ( z ) satisfies
f (^) ( α ) = 0
To find the zeros of
sinh
(^) z , we observe that
sinh
(^) z
= 0
(^) sinh
(^) z | = 0
(^) sinh
2 x (^) + sin
2 y
= 0
Hence,
x
and
y
must satisfy sinh
(^) x
= 0 and sin
(^) y
= 0, thus giving
x
= 0 and
y
=
kπ
k
is any integer.
The zeros of sinh
(^) z
are
z
kπi
k
is any integer.
13
Consider the complex sine functionMapping properties of the complex sine function
w
= sin
(^) z
= sin
(^) x (^) cosh
(^) y (^) +
(^) i (^) cos
(^) x (^) sinh
(^) y,
z
x (^) +
(^) iy,
suppose we write
w
=
u (^) +
(^) iv
, then
u
= sin
(^) x (^) cosh
(^) y
and
v
= cos
(^) x (^) sinh
(^) y.
We find the images of the coordinates lines
x
=
α
and
y
=
β
. When
x
=
α , u
= sin
(^) α
(^) cosh
(^) y
and
v
= cos
(^) α
(^) sinh
(^) y
. By eliminating
y
in the
above equations, we obtain
u 2
sin
2 α^
−
v 2
cos
2 α^
= 1
which represents a hyperbola in the
w -plane.
14
Take
α >
When 0
< α <
2 π , u
= sin
(^) α
(^) cosh
(^) y >
0 for all values of
y , so the line
x
=
α
is mapped onto the right-hand branch of the
hyperbola. Likewise, the line
x
=
α
is mapped onto the left-hand
In particular, whenbranch of the same hyperbola.
α
=
2 π , the line
x
=
2 π
is mapped onto the line
segment
v
= 0,
u
≥
1 (degenerate hyperbola).
Also, the
y -axis is
mapped onto the
v -axis.
We conclude that the infinite strip
{ 0
≤
Re
z
≤
2 π }
is mapped to
the right half-plane
{ u ≥ 0 }.
By symmetry, the other infinite strip
{ −
2 π
≤
Re
z
≤
0 }
is mapped to the left half-plane
{ u ≤ 0 }.
16
Consider the image of a horizontal line
y
=
β
( β >
2 π
≤
x
≤
2 π
under the mapping
w
= sin
(^) z .
When
y
=
β , u
= sin
(^) x (^) cosh
(^) β
and
v
= cos
(^) x (^) sinh
(^) β .
By eliminating
x
in the above equations, we obtain
u 2
cosh
2 β^
v 2
sinh
2 β^
= 1
which represents an ellipse in the
w -plane.
The upper line
y
=
β
(the lower line
y
=
β ) is mapped onto
the upper (lower) portion of the ellipse.
When
β
the line segment
y
= 0
2 π
≤
x
≤
2 π
is mapped
onto the line segment
v
= 0
u
≤
1, which is a degenerate
ellipse.
17
(a) Find the general solution forExample
e z
= 1 +
(^) i .
Is
e z | bounded when
Re (
z ) =
β ?
(b) Show that
(^) sin
(^) z | is bounded when Im (
z ) =
(^) α
.
(a) Solution
e z =
e x (cos
(^) y (^) +
(^) i (^) sin
(^) y ) = 1 +
(^) i, z
x (^) +
(^) iy
e x
=
i | =
2 so that
x
=
ln 2; tan
(^) y
= 1 so that
y
=
4 π
kπ, k
is integer.
| e z | = e x = e β
so that
e z | is bounded when Re
z
β .
(b)
(^) sin
(^) z | 2
= sin
2 x^ (^) cosh
2 y^
2 x^ (^) sinh
2 y^
= sin
2 x^ (^) + sinh
2 y^ .
When
Im
z
=
α,
(^) | (^) sin(
x (^) +
(^) iα
2
= sin
2 x^ (^) + sinh
2 α^
≤
1 + sinh
2 α^ .
Hence,
(^) sin
(^) z | ≤
√
1 + sinh
2 α^
when Im
z
(^) α
.
19
Suppose a functionReflection principle
f
is analytic in a domain
D
which includes part of
the real axis and
D
is symmetric about the real axis. The reflection
f principle states that (^) ( z ) =
f (^) ( z ) if and only if
f (^) ( z ) is real when
z
is real.
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