**Hyperbolic Functions: Algebra with sinh x, cosh x, e^x**

**sinh x and cosh x defined in terms of e^x**

sinh
x = ( e^x - e^-x) / 2 (I)

cosh
x = ( e^x + e^-x) / 2 (II)

x can be real
or complex

**sinh^2 x and cosh^2 x**

sinh^ 2 x

= (sinh x)^2

= (( e^x -
e^-x) / 2)^2

= 1/4 *
((e^x)^2 – 2*e^x*e^-x + (e^-x)^2)

=
1/4 * ((e^x)^2 – 2 + (e^-x)^2) (III)

cosh^ 2 x

= (cosh x)^2

= (( e^x +
e^-x) / 2)^2

= 1/4 *
((e^x)^2 + 2*e^x*e^-x + (e^-x)^2)

=
1/4 * ((e^x)^2 + 2 + (e^-x)^2) (IV)

**Product of sinh x and cosh x**

sinh x * cosh x

= (e^x – e^-x)/2
* (e^x + e^-x)/2

= 1/4 * ((e^x –
e^-x) * (e^x + e^-x))

= 1/4 * (
e^x*e^x + e^x*e^-x – e^-x*e^x – e^-x*e^-x )

=
1/4 * ( e^(2*x) – e^(-2*x) ) (V)

Note that e^x *
e^-x = 1.

**Sums and Differences with sinh^2 x and cosh^2 x**

sinh^2 x +
cosh^2 x

= 1/4 * (
(e^x)^2 – 2 (e^-x)^2 ) + 1/ 4 * ( (e^x)^2 + 2 + (e^-x)^2 )

= 1/4 * ( 2 *
(e^x)^2 + 2 * (e^x)^-2 )

=
1/2 * ( (e^x)^2 + (e^-x)^2 ) (VI)

sinh^2 x -
cosh^2 x

= 1/4 * (
(e^x)^2 – 2 (e^-x)^2 ) - 1/ 4 * ( (e^x)^2 + 2 + (e^-x)^2 )

= 1/4 * ( -4 )

=
-1 (VII)

This implies
that cosh^2 x – sinh^2 x = 1.

**Binomial expansions of involving sums and differences of e^x and e^-x**

(e^x
+ e^-x)^2 = (e^x)^2 + 2 + (e^-x)^2

(e^x
– e^-x)^2 = (e^x)^2 – 2 + (e^-x)^2

(
(e^x)^2 + (e^-x)^2 )^2 = (e^x)^4 + 2 +
(e^-x)^4

(
(e^x)^2 – (e^-x)^2 )^2 = (e^x)^4 – 2 + (e^-x)^4 (VIII)

**Product of sinh^2 x and cosh^2 x**

Substitutions: α = e^x and β = e^-x, therefore α*β = e^x *
e^-x = 1

sinh^2 x *
cosh^2 x

= 1 /4 * (α^2 –
2 + β^2) * 1 /4 * (α^2 + 2 + β^2)

= 1/16 * (α^4 +
2*α^2 + α^2*β^2 – 2*α^2 – 4 – 2*β^2 + α^2*β^2 + 2*β^2 + β^4)

= 1/16 * (2*α^2*β^2
+ α^4 + β^4 – 4)

Back substitute:

= 1/16 * (2 +
(e^x)^4 + (e^-x)^4 – 4)

= 1/16 * (
(e^x)^4 – 2 + (e^-x)^4 )

=
1/16 * ( (e^x)^2 – (e^-x)^2 ) ^2 (IX)

See you in
March! I am working on coordinate
conversions of very unusual (at least in my opinion) coordinate systems, more
programs, and hopefully making sense of basic tensor calculus.

Eddie

This blog is
property of Edward Shore, 2017

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