Lecture 21 Hyperbolic Functions

Lecture 21:
Hyperbolic Functions
Dan Sloughter
Furman University
Mathematics 39
April 8, 2004
21.1 Hyperbolic sine and cosine
Definition 21.1. For any z ∈ C, we define the hyperbolic sine function by
and the hyperbolic cosine function by
Proposition 21.1. For any z ∈ C,
sinh(z) = ez − e −z
2
cosh(z) = ez + e −z
.
2
d
sinh(z) = cosh(z)
dz
and
d
cosh(z) = sinh(z).
dz
Proof. We have
d
dz sinh(z) = d ( ) e z − e −z
= ez + e −z
= cosh(z)
dz 2
2
and
d
dz cosh(z) = d ( ) e z + e −z
= ez − e −z
= sinh(z).
dz 2
2
1

Note that
cos(z) = cosh(iz) and sin(z) = −i sinh(iz)
and
It follows that
cosh(z) = cos(iz) and sinh(z) = −i sin(iz).
and
sinh(−z) = −i sin(−iz) = i sin(iz) = − sinh(z),
cosh(−z) = cos(−iz) = cos(iz) = cosh(z),
cosh 2 (z) − sinh 2 (z) = cos 2 (iz) − (−i sin(iz)) 2 = cos 2 (iz) + sin 2 (iz) = 1,
sinh(z 1 + z 2 ) = −i sin(i(z 1 + z 2 )
= −i sin(iz 1 ) cos(iz 2 ) − i cos(iz 1 ) sin(iz 2 )
= sinh(z 1 ) cosh(z 2 ) + cosh(z 1 ) sinh(z 2 ),
cosh(z 1 + z 2 ) = cos(i(z 1 + z 2 ))
= cos(iz 1 ) cos(iz 2 ) − sin(iz 1 ) sin(iz 2 )
= cosh(z 1 ) cosh(z 2 ) + sinh(z 1 ) sinh(z 2 ).
Moreover, if z = x + iy, then iz = −y + ix, and we have
and
sinh(z) = −i sin(iz)
= −i(sin(−y) cos(ix) + cos(−y) sin(ix)
= sinh(x) cos(y) + cosh(x) sin(y),
cosh(z) = cos(iz)
= cos(−y) cos(ix) − sin(−y) sin(ix)
= cosh(x) cos(y) + i sinh(x) sin(y),
| sinh(z)| 2 = | − i sin(iz)| 2 = sin 2 (−y) + sinh 2 (x) = sinh 2 (x) + sin 2 (y)
| cosh(z)| 2 = | cosh(iz)| 2 = cos 2 (−y) + sinh 2 (x) = sinh 2 (x) + cos 2 (y).
Also note that it follows that both sinh(z) and cosh(z) are periodic with
period 2πi, that sinh(z) = 0 if and only if z = nπi, n = 0, ±1, ±2, . . ., and
cosh(z) = 0 if and only if z = i ( π
2 + nπ) , n = 0, ±1, ±2, . . ..
2

21.2 The other hyperbolic functions
In analogy with the the circular trigonometric functions, we also define
and
tanh(z) = sinh(z)
cosh(z) ,
coth(z) = cosh(z)
sinh(z) ,
sech(z) =
csch(z) =
1
cosh(z) ,
1
sech(z) .
These are all analytic in their domains of definition. One may show, as in
the real-variable case, that
and
d
dz tanh(z) = sech2 (z),
d
dz coth(z) = − csch2 (z),
d
sech(z) = − sech(z) tanh(z),
dz
d
csch(z) = − csch(z) coth(z).
dz
3