Complex
Analysis. FreeTutorial
The hyperbolic cosine and
hyperbolic sine functions are:
Sinh(z)
=
Cosh(z)
=
We can easily create the
other complex hyperbolic trigonometric
functions.
tanh(z)
=
coth(z)
=
sech(z)
=
csch(z)
=
The derivatives of the hyperbolic
functions are:
Sinh(z)
= Cosh(z)
Cosh(z)
= Sinh(z)
tanh(z)
= sech(z)^{2}
coth(z)
=  csch(z)^{2}
sech(z)
=  sech(z)tanh(z)
csch(z)
=  csch(z)coth(z)
The hyperbolic cosine and hyperbolic sine
can be expressed as:
csch(z)
= csch(x+iy)
= = Cos(y)csch(x)
+ iSinh(x)Sin(y)
Sinh(z)
= Sinh(x+iy)
=
= Sinh(x)Cos(y)
+ iCosh(x)Sin(y)
Some of the important identities
involving the hyperbolic functions are:
Cosh(z)^{2}
 Sinh(z)^{2}
= 1
Sinh(z_{1}+
z_{2})
= Sinh(z_{1})Cosh(z_{2})
+ Cosh(z_{1})Sinh(z_{2})
Cosh(z_{1}+
z_{2})
= Cosh(z_{1})Cosh
(z_{2})
+ Sinh(z_{1})Sinh(z_{2})
Sinh(z
+)
= Sinh(z)
Cosh(z
+)
= Cosh(z)
Cosh(z)
= Cosh(z)
Sinh(z)
= Sinh(z)
There is a connection between
complex hyperbolic and complex trigonometric
functions:
Cosh(z)
= Cos(iz)
Sinh(z)
=  iCos(iz)
Cos(z)
= Cosh(iz)
Sin(z)
=  iSinh(iz)
Sinh(iz)
= iSin(z)
Sin(iz)
= iSinh(z)
by
Tetyana Butler
