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
