Math334  Practice Exercises 4
1
Jan 31 and Feb 2
1. By using sin
z
=
e
iz

e

iz
2
i
, etc, show the followings
(a) sin
iz
=
i
sinh
z
(b) sinh
iz
=
i
sin
z
(c) cos
iz
= cosh
z
(d) cosh
iz
= cos
z
2. By using sin
z
=
e
iz

e

iz
2
i
, etc, show the followings
(a) sin(
z
+
w
) = sin
z
cos
w
+ sin
w
cos
z
.
(b) cos(
z
+
w
) = cos
z
cos
w

sin
z
sin
w
.
(c) sinh(
z
+
w
) = sinh
z
cosh
w
+ sinh
w
cosh
z
.
(b) cosh(
z
+
w
) = cosh
z
cosh
w
+ sinh
z
sinh
w
.
3. By using sin
z
=
e
iz

e

iz
2
i
, etc, show the followings
(a) sin
2
z
+ cos
2
z
= 1
(b) cosh
2
z

sinh
2
z
= 1
(c) (sin
z
)
0
= cos
z
(d) (cos
z
)
0
=

sin
z
(e) (sinh
z
)
0
= cosh
z
(f) (cosh
z
)
0
= sinh
z
4. Solve cosh
x
= 2 for
x
∈
R
.
5. Solve cosh
z
= 2 for
z
∈
R
.
6. Solve sin
z
= 1.
7. Solve
e
z
= 4 +
i
.
8. Solve cos
z
= 1 +
i
.
9. Considering the notes for Feb 2, we tried to analyze how the function
f
(
z
) = sin
z
behaves.
Let
S
=
{
a
+
yi
}
so that
a
is fixed and
y
is arbitrary. What is
f
(
S
)? (Recall sin(
x
+
iy
) = sin
x
cosh
y
+
i
cos
x
sinh
y
and cosh
2
y

sinh
2
y
= 1)
1
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Math334  Practice Exercises 5
1
Logarithms
1. Evaluate the followings
(a) log 2
(b) Log (3 + 2
i
)
(c) Log (

3 + 2)
2. While we know as multivalued functions, log
z
+log
w
= log(
zw
), give an example so that Log
z
+Log
w
6
=
Log (
zw
).
3. Which of the following statements are true? Why?
(a)
e
Log
z
=
z
(b) Log
e
z
=
z
(c)
e
log
z
=
z
(d) log
e
z
=
z
4. Solve
e
2
iz
+ 3
e
iz
+ 2 = 0 for
z
.
5. What is the branch cut of Log (
z
2
+ 3)?
*6. We know that the branch cut of the principal branch of log
z
, i.e., Log
z
is the negative reals. Try to
come up with an equation involving Log .
(a) A branch of log
z
with the positive reals as branch cut, and takes value of
iπ
at
z
=

1.
(Hint:
Compute some examples of Log (

z
) and compare to Log
z
.)
(b) A branch of log
z
with positive imaginary axis as branch cut, and takes value of 0 at
z
= 1.
(c) A branch of log
z
with negative imaginary axis as branch cut, and takes value of 0 at
z
= 1.
2
Complex Powers
1. Evaluate the followings
(a)
i
1
/
2
(b)
i
i