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1. The Hyperbolic Function

,sinh , cosh

2 2

z z z z

e e e e

z z z

− −

− +

∀ ∈ = = .

Furthermore,

sinh tanh

cosh z z

z

=

,

sinh cosh

d

z z

dz = _,

cosh coth

sinh z z

z

=

,

cosh sinh

d

z z

dz = ,

1 sec

cosh hz

z

= 2

(tanh ) sec

d

z h z

dz = ,

z hz

sinh 1

csc = d (sechz) sechztanz

dz = − _.

The relationship between hyperbolic function and trigonometric function: a. cosh

( )

iz =cosz c. −isinh

( )

iz =sinz

b. cos

( )

iz =coshz d. −isin

( )

iz =sinhz

Some of the most frequently used identities involving hyperbolic sine and cosine functions are

(

)

sinh −z = −sinhz

, sinhz=sinh cosx y+icosh sinx y, ∀ =z x iy+ ,

(

)

cosh −z =coshz

, coshz=cosh cosx y+isinh sinx y, ∀ =z x iy+ ,

2 2

cosh z−sinh z=1_, 2 2 2

sinhz =sinh x+sin y, ∀ =z x iy+ _,

(

1 2

)

1 2 1 2

sinh z +z =sinhz coshz +coshz sinhz

,

2 _{2 2}

coshz =sinh x+cos y, ∀ =z x iy+ _,

(

1 2

)

1 2 1 2

cosh z +z =coshz coshz +sinhz sinhz

.

(

)

logz=lnr+i θ+2nπ ,n∈ .

The principal value of logzis denoted by Log z, define Log z=lnr+iθ,−

π

<

θ

≤

π

.

(

)

log 1 ln 2 2 , 0, 1, 2,

4

i i π nπ n

+ = + + = ± ± and the principal value is

(

1

)

ln 2

4

Log +i = +i π .

Let z z z, ,_{1 2}∈

a. d logz 1

dz = z c. log

(

z z1 2

)

=logz1+logz2

b. elogz =z d. 1

1 2

2

log z logz logz

z = −

Exercises:

1. Show that sin

( )

iz =isinhz

2. Find all values of z for which the following equations hold

a. sinh

2 i

z= b. coshz=1

3. Show that sinh

(

z i+

π

)

= −sinhz

4. Find the principal value for the following

a. Log ie

( )

2 b. Log i

(

2− 2

)

5. Find all the values of logzfor the log

(

−3

)

6. Find all the values of z for which the following equations hold:

a. 1

4 i