Sunday, February 26, 2017

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



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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