Física Matemática I - Funções de Variável Complexa

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(
F
´
ısica
 
 
Mate
m
´
atica
 
 
I
 
 
-
 
 
F
IS01207
 
 
-
 
 
Unidade
 
 
I
Lista
 
 
3
 
 
-
 
 
F
un
¸
c
˜
oes
 
 
d
e
 
 
V
ar
i
´
a
v
e
l
 
 
Complexa
 
 
-
 
 
FUN
C
¸
 
O
˜
E
S
 
 
E
L
E
M
E
N
T
A
RES
) (
1.
) (
Determine todos os valores de 
z
 
tais que
(
a
)
 
e
z
 
=
 
−
2; 
 
(
b
)
 
e
z
 
=
 
1
 
+
 
i
√
3;
) (
e
(2
z
−
1)
 
=
 
1
) (
Resp:
 
(a)
 
z
 
=
 
log
 
2
 
+
 
i
(2
n
 
+
 
1)
π
;
 
(b)
 
z
 
=
 
log
 
2
 
+
 
iπ
(2
n
 
+
 
1
/
3);
 
(c)
 
z
 
=
 
(1
/
2)
 
+
 
inπ
;
 
com
 
n
 
inteiro
Determine
 
a
 
A
 
e
 
φ
 
em:
sin(
ωt
)
 
+
 
sin(
ωt
 
+
 
60
o
)
 
+
 
sin(
ωt
 
−
 
30
o
)
 
=
 
A
 
sin(
ωt
 
+
 
φ
)
.
Resp:
 
A
 
=
 
2
,
 
4
 
;
 
φ
 
=
 
8
,
 
8
o
N
u
m
 
 
circu
i
to
 
 
RLC
 
 
e
m
 
 
s
´
er
i
e
 
 
liga
d
o
 
 
a
 
 
u
m
a
 
 
fo
n
te
 
 
alter
n
ada
 
 
de
 
 
am
p
lit
u
de
 
 
V
0
 
 
e
 
 
f
req
u
¨
ˆ
encia
 
 
ω
,
 
 
a
 
 
carga
 
 
no 
capacitor
 
q
(
t
)
 
o
b
e
d
e
ce
 
a
 
e
q
u
a
¸
c
˜
ao
 
d
iferencial
) (
2.
) (
3.
) (
d
2
q
L
dt
2
) (
dq
) (
q
) (
+
 
R
+
= 
V
0
 
cos(
ωt
)
.
 
dt
C
) (
(a)
 
S
u
bstit
u
i
n
do
 
c
os(
ω
t
)
 
p
or
 
e
i
ω
t
 
na
 
equa
¸
c
˜
ao
 
ac
i
m
a,
 
mos
t
re
 
q
u
e
 
a
 
s
o
l
u
¸
c
˜
ao
 
estac
io
n
´
ar
i
a
 
com
p
lexa
 
´
e
) (
V
0
 
e
iωt
1
/C
 
−
 
ω
2
L
 
+
 
iωR
) (
q
(
t
)
 
=
) (
.
) (
(
b
)
 
Ut
i
lizando
 
o
 
r
e
su
l
tado
 
ac
ima
 
e
 
o
 
fato
 
de
 
q
ue
 
a
 
cor
r
e
n
te
 
´
e
 
a
 
der
i
v
a
d
a
 
tem
p
o
r
al
 
d
a
 
carga,
 
m
ostre
 
q
u
e
 
a 
corre
n
te
 
e
stac
i
on
´
ar
i
a
 
r
e
al
 
n
o
 
circ
u
ito
 
´
e
) (
V
) (
0
) (
I
(
) (
t
)
 
=
cos(
) (
ωt
 
−
 
θ
)
) (
,
) (
Z
) (
com
) (
 
) (
 
) (
ωL
 
−
 
1
) (
Z
 
=
 
R
2
 
+
 
(
ωL
 
− 
1
/ωC
)
2
 
1
/
2 
 
e
 
θ
 
=
 
arctan
) (
/ωC
) (
,
) (
R
) (
rep
r
e
se
n
tan
d
o
 
a
 
im
p
e
d
ˆ
ac
i
a
 
e
 
o
 
at
r
as
o
 
de
 
fase
 
da
 
corre
n
te,
 
res
p
ec
t
i
v
ame
n
te 
Mostre
 
que
 
|
 
sin
 
z
|
 
≥
 
|
 
sin
 
x
|
 
e
 
|
 
cos
 
z
|
 
≥
 
|
 
cos
 
x
|
.
Determine
 
t
o
d
as
 
as
 
r
a
´
ı
z
es
 
d
a
s
 
e
q
ua
¸
c
˜
oe
s:
(
a
)
 
cos
 
z
 
=
 
2;
 
(
b
)
 
cosh
 
z
 
=
 
1
/
2;
 
(
c
)
 
sinh
 
z
 
=
 
i
) (
4.
5.
) (
Resp:
 
(a)
 
z
 
=
 
2
nπ
 
+
 
i
 
arccosh
 
2;
 
(b)
 
z
 
=
 
iπ
(2
n
 
±
 
π/
3);
 
(c)
 
z
 
=
 
iπ
(2
n
 
+ 1
/
2);
 
com
 
n
 
inteiro.
Quando
 
n 
=
 
0
,
 
1
,
 
2
,
 
...
,
 
mostre
 
que
) (
6.
) (
1
) (
1
) (
(
a
)
 
log
 
1
 
=
 
±
2
nπi
; 
 
(
b
)
 
log
(
−
1)
 
=
 
±
(2
n
 
+
 
1)
πi
; 
 
(
) (
)
 
log
 
i
 
=
πi
 
±
 
2
) (
; 
 
(
) (
d
)
) (
log
(
) (
)
 
=
πi
 
± 
nπi
) (
1
/
2
) (
c
) (
nπi
) (
i
) (
2
) (
4
) (
7.
) (
Mostre
 
que
) (
1
) (
1
) (
1
) (
(
) (
)
 
Log
 
(
) (
)
 
=
 
1
 
− 
 
πi
) (
;
) (
(
) (
b
) 
Log
 
(1
 
−
 
i
) (
)
 
=
Log
 
2
 
− 
 
πi
) (
a
) (
−
) (
ei
) (
2
) (
2
) (
4
) (
8.
) (
Quando
 
n
 
=
 
0
,
 
1
,
 
2
,
 
...
,
 
mostre
 
que
) (
 
) (
 
) (
 
) (
 
) (
1
) (
1
) (
(1
 
+
 
i
) (
) 
 
=
 
exp 
 
− 
 
π
 
±
 
2
) (
i
) (
nπ
exp
iLog
 
2
) (
4
) (
2
)