Before we start on the gallery we define the term “entire function”.
Definition.
A function that is analytic at every point in the complex plane is called an
entire function
. We will see that e
z
,
z
n
, sin(
z
) are all entire functions.
2.8.1
Gallery of functions, derivatives and properties
The following is a concise list of a number of functions and their derivatives. None of the
derivatives will surprise you. We also give important properties for some of the functions.
The proofs for each follow below.
1.
f
(
z
) = e
z
= e
x
cos(
y
) +
i
e
x
sin(
y
).
Domain = all of
C
(
f
is entire).
f
0
(
z
) = e
z
.
2.
f
(
z
)
≡
c
(constant)
Domain = all of
C
(
f
is entire).
f
0
(
z
) = 0
.
3.
f
(
z
) =
z
n
(
n
an integer
≥
0)
Domain = all of
C
(
f
is entire).
f
0
(
z
) =
nz
n

1
.
4.
P
(
z
) (polynomial)
A polynomial has the form
P
(
z
) =
a
n
z
n
+
a
n

1
z
n

1
+
. . .
+
a
0
.
Domain = all of
C
(
P
(
z
) is entire).
P
0
(
z
) =
na
n
z
n

1
+ (
n

1)
a
n

1
z
n

1
+
. . .
+ 2
a
2
z
+
a
1
.
5.
f
(
z
) = 1
/z
Domain =
C
 {
0
}
(
the punctured plane
).
f
0
(
z
) =

1
/z
2
.
6.
f
(
z
) =
P
(
z
)
/Q
(
z
) (rational function).
When
P
and
Q
are polynomials
P
(
z
)
/Q
(
z
) is called a rational function.
If we assume that
P
and
Q
have no common roots, then:
Domain =
C
 {
roots of
Q
}
f
0
(
z
) =
P
0
Q

PQ
0
Q
2
.
2
ANALYTIC FUNCTIONS
13
7. sin(
z
), cos(
z
)
Definition.
cos(
z
) =
e
iz
+ e

iz
2
,
sin(
z
) =
e
iz

e

iz
2
i
(By Euler’s formula we know this is consistent with cos(
x
) and sin(
x
) when
z
=
x
is
real.)
Domain: these functions are entire.
d
cos(
z
)
dz
=

sin(
z
)
,
d
sin(
z
)
dz
= cos(
z
)
.
Other key properties of sin and cos:
 cos
2
(
z
) + sin
2
(
z
) = 1
 e
z
= cos(
z
) +
i
sin(
z
)
 Periodic in
x
with period 2
π
, e.g. sin(
x
+ 2
π
+
iy
) = sin(
x
+
iy
).
 They are not bounded!
 In the form
f
(
z
) =
u
(
x, y
) +
iv
(
x, y
) we have
cos(
z
) = cos(
x
) cosh(
y
) +
i
sin(
x
) sinh(
y
)
sin(
z
) = sin(
x
) cosh(
y
) +
i
cos(
x
) sinh(
y
)
(cosh and sinh are defined below.)
 The zeros of sin(
z
) are
z
=
nπ
for
n
any integer.
The zeros of cos(
z
) are
z
=
π/
2 +
nπ
for
n
any integer.
(That is, they have only real zeros that you learned about in your trig. class.)
8. Other trig functions cot(
z
), sec(
z
) etc.
Definition.
The same as for the real versions of these function, e.g.
cot(
z
) =
cos(
z
)
/
sin(
z
), sec(
z
) = 1
/
cos(
z
).
Domain: The entire plane minus the zeros of the denominator.
Derivative: Compute using the quotient rule, e.g.
d
tan(
z
)
dz
=
d
dz
sin(
z
)
cos(
z
)
=
cos(
z
) cos(
z
)

sin(
z
)(

sin(
z
))
cos
2
(
z
)
=
1
cos
2
(
z
)
= sec
2
z
(No surprises there!)
9. sinh(
z
), cosh(
z
) (
hyperbolic sine and cosine
)
Definition.
cosh(
z
) =
e
z
+ e

z
2
,
sin(
z
) =
e
z

e

z
2
Domain: these functions are entire.
d
cosh(
z
)
dz
= sinh(
z
)
,
d
sinh(
z
)
dz
= cosh(
z
)
Other key properties of cosh and sinh:
2
ANALYTIC FUNCTIONS
14
 cosh
2
(
z
)

sinh
2
(
z
) = 1
 For real
x
, cosh(
x
) is real and positive, sinh(
x
) is real.
 cosh(
iz
) = cos(
z
),
sinh(
z
) =

i
sin(
iz
).
10. log(
z
) (See Topic 1.)
Definition.
log(
z
) = log(

z

) +
i
arg(
z
).
Branch: Any branch of arg(
z
).
Domain:
C
minus a branch cut where the chosen branch of arg(
z
) is discontinuous.
d
dz
log(
z
) =
1
z
11.
z
a
(any complex
a
) (See Topic 1.)
Definition.
z
a
= e
a
log(
z
)
.
Branch: Any branch of log(
z
).
Domain: Generally the domain is
C
minus a branch cut of log. If
a
is an integer
≥
0
then
z
a
is entire. If
a
is a negative integer then
z
a
is defined and analytic on
C
=
{
0
}
.
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 Spring '17
 Derivative, lim, 2Z