MEETING: 21-22
Learning Objectives: Students are able to
1. explain hyperbolic functions
2. explain logarithmic functions
1.
The Hyperbolic Function
Definition:
∀z ∈ ,sinh z =
e z − e− z
e z + e− z
, cosh z =
.
2
2
Furthermore,
sinh z
d
sinh z = cosh z
cosh z ,
dz
,
cosh z
d
coth z =
cosh z = sinh z
sinh z ,
dz
,
1
d
sec hz =
(tanh z ) = sec h 2 z
cosh z ,
dz
,
1
d
csc hz =
(sec hz ) = − sec hz tan z
sinh z ,
dz
.
The relationship between hyperbolic function and trigonometric function:
tanh z =
a. cosh ( iz ) = cos z
c. −i sinh ( iz ) = sin z
b. cos ( iz ) = cosh z
d. −i sin ( iz ) = sinh z
Some of the most frequently used identities involving hyperbolic sine and cosine functions are
sinh ( − z ) = − sinh z
sinh z = sinh x cos y + i cosh x sin y , ∀z = x + iy ,
,
cosh ( − z ) = cosh z
cosh z = cosh x cos y + i sinh x sin y , ∀z = x + iy ,
,
2
2
cosh z − sinh z = 1 ,
2
sinh z = sinh 2 x + sin 2 y , ∀z = x + iy ,
sinh ( z1 + z 2 ) = sinh z1 cosh z 2 + cosh z1 sinh z 2
,
cosh ( z1 + z 2 ) = cosh z1 cosh z 2 + sinh z1 sinh z2
.
2.
The Logarithmic Function
For every z = reiθ ,
2
cosh z = sinh 2 x + cos 2 y , ∀z = x + iy ,
log z = ln r + i (θ + 2nπ ) , n ∈ .
The principal value of log z is denoted by Log z , define Log z = ln r + iθ , − π < θ ≤ π .
Example
π

log (1 + i ) = ln 2 + i  + 2nπ  , n = 0, ±1, ±2,…
4

and
the
principal
π 
Log (1 + i ) = ln 2 + i   .
4
Properties:
Let z , z1 , z2 ∈ a.
1
d
log z =
dz
z
b. elog z = z
c. log ( z1 z2 ) = log z1 + log z2
z 
d. log  1  = log z1 − log z2
 z2 
Exercises:
1.
Show that sin ( iz ) = i sinh z
2.
3.
Find all values of z for which the following equations hold
i
a. sinh z =
b. cosh z = 1
2
Show that sinh ( z + iπ ) = − sinh z
4.
Find the principal value for the following
a. Log ( ie 2 )
(
b. Log i 2 − 2
)
5.
Find all the values of log z for the log ( −3)
6.
Find all the values of z for which the following equations hold:
iπ
a. Log z = 1 −
b. e z = −ie
4
value
is